#494482

order	512
nilpotency class	4
isoclinism	isoclinic class of #33657
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#481
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1874
prime radical	#60450	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1385 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280077, #72280321, #72280998, …
maximal normal subgroup of	#72280077, #72280321, #72280998, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401678);