#511764

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

3069 conjugacy classes of subgroups; only the 220 normal subgroups are shown.

normal subgroups only

This group has 3069 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:

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

Polycyclic presentation:

2^9|b^2=i,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	#72320293, #72343955, #72343961
maximal normal subgroup of	#72320293, #72343955, #72343961
maximal quotient of	#72320606, #72320641

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418960);