#373971

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

838 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(2,3)(5,6)(7,9)(8,10)(11,12)(13,14,18,23,19,24,17,22)(15,20,26,34,27,35,31,39)(16,21,30,38,29,37,32,40)(25,33,42,44,41,43,28,36);(1,2,4,3)(5,7,8,11)(6,9,10,12)(13,15)(14,20)(16,28)(17,31)(18,26)(19,27)(21,36)(22,39)(23,34)(24,35)(25,30)(29,42)(32,41)(33,38)(37,44)(40,43);(6,10)(9,12)(13,16,18,30,19,29,17,32)(14,21,23,38,24,37,22,40)(15,25,26,42,27,41,31,28)(20,33,34,44,35,43,39,36);(1,3,4,2)(16,29)(21,37)(25,41)(28,42)(30,32)(33,43)(36,44)(38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=h,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281167);