#358985

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

986 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,d^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,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, 266181);