#335216

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	#25631	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

783 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)(19,26)(20,22)(21,25)(23,28)(24,29)(27,32)(30,31);(1,3,4,8)(2,5,6,7)(10,14)(12,15)(13,16)(19,27)(23,31)(26,32)(28,30);(9,10)(11,13)(12,16)(14,15)(18,22)(20,29)(23,26)(24,25)(28,30)(31,32);(2,6)(3,8)(17,19,21,27)(18,23,25,31)(20,30,29,28)(22,26,24,32)

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=h,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, 242412);