#495131

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	#60643	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1272 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(2,4)(5,6,10,16,12,17,11,32)(7,18,21,43,23,44,22,51)(8,14,26,40,24,38,27,56)(9,28,29,34,31,57,30,15)(13,33,36,50,37,49,42,20)(19,41,47,64,45,62,48,67)(25,53,54,59,52,68,55,39)(35,58,61,65,60,66,63,46);(5,7,9,20,10,21,29,33,12,23,31,50,11,22,30,49)(6,13,34,44,32,42,28,43,17,37,15,18,16,36,57,51)(8,19,25,46,26,47,54,58,24,45,52,65,27,48,55,66)(14,35,59,62,56,63,53,64,38,60,39,41,40,61,68,67);(1,2,3,4)(5,8)(6,14)(7,19)(9,25)(10,26)(11,27)(12,24)(13,35)(15,39)(16,40)(17,38)(18,41)(20,46)(21,47)(22,48)(23,45)(28,53)(29,54)(30,55)(31,52)(32,56)(33,58)(34,59)(36,61)(37,60)(42,63)(43,64)(44,62)(49,66)(50,65)(51,67)(57,68);(8,24)(14,38)(19,45)(25,52)(26,27)(35,60)(39,59)(40,56)(41,62)(46,65)(47,48)(53,68)(54,55)(58,66)(61,63)(64,67)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402327);