#511763

order	512
nilpotency class	4
isoclinism	isoclinic class of #511763
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32878
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3573 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3118 conjugacy classes of subgroups; only the 220 normal subgroups are shown.

normal subgroups only

This group has 3118 conjugacy classes of subgroups — too many to draw. Only its 220 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,9)(4,10)(5,8)(6,24)(7,13)(11,21)(12,22)(14,36)(15,25)(16,31)(17,43)(18,26)(19,34)(20,47)(23,28)(27,48)(29,38)(30,40)(32,41)(33,44)(35,45)(37,56)(39,55)(42,50)(46,52)(49,61)(51,62)(53,63)(54,64)(57,59)(58,60);(2,8)(5,21)(6,22)(9,25)(10,26)(11,12)(13,28)(15,40)(17,41)(18,44)(20,45)(23,48)(24,27)(29,49)(30,32)(31,50)(33,35)(34,52)(36,56)(37,59)(39,60)(42,62)(43,51)(46,63)(47,53)(54,55)(57,58)(61,64);(1,3)(2,9)(4,14)(5,15)(6,17)(7,16)(8,25)(10,29)(11,30)(12,32)(13,31)(18,37)(19,38)(20,39)(21,40)(22,41)(23,42)(24,43)(26,49)(27,51)(28,50)(33,54)(34,36)(35,55)(44,59)(45,60)(46,61)(47,58)(48,62)(52,56)(53,57)(63,64);(1,4)(2,10)(3,14)(5,18)(6,20)(7,19)(8,26)(9,29)(11,33)(12,35)(13,34)(15,37)(16,38)(17,39)(21,44)(22,45)(23,46)(24,47)(25,49)(27,53)(28,52)(30,54)(31,36)(32,55)(40,59)(41,60)(42,61)(43,58)(48,63)(50,56)(51,57)(62,64)

Polycyclic presentation:

2^9|e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,h^a=hi,e^b=eh,h^b=hi,g^c=gi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281969, #72320293, #72343953, …
maximal normal subgroup of	#72281969, #72320293, #72343953, …
maximal quotient of	#72320508, #72320532

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418959);