#363728

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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	C₂²×C₄⋊C₄
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1136 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=f,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di

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, 270924);