#335233

order	512
nilpotency class	3
isoclinism	isoclinic class of #331791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

701 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,4,6)(3,7,8,5)(10,13)(14,16)(17,18,21,25)(19,22,26,31)(20,23,28,24)(27,29,32,30);(1,3,4,8)(2,5,6,7)(10,14)(12,15)(13,16)(17,19)(18,22)(20,27)(21,26)(23,29)(24,30)(25,31)(28,32);(9,10,11,13)(12,16,15,14)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(2,6)(3,8)(18,23)(19,26)(20,28)(22,30)(24,25)(29,31)

Polycyclic presentation:

2^9|a^2=eh,b^2=e,c^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,f^a=fh,c^b=cg,d^b=deh,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 242429);