#511765

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
prime radical	#63091	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3573 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3104 conjugacy classes of subgroups; only the 220 normal subgroups are shown.

normal subgroups only

This group has 3104 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:

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

Polycyclic presentation:

2^9|a^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	#72281969, #72343955, #72343960
maximal normal subgroup of	#72281969, #72343955, #72343960

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418961);