#96296

order	512
nilpotency class	3
isoclinism	isoclinic class of #7694
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

630 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=hi,d^2=g,f^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fgh,c^b=cgh,d^b=dh

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, 3492);