#493172

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

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

762 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=e,g^2=h,h^2=i,b^a=be,c^a=ci,d^a=di,d^c=di

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400368);