#335601

order	512
nilpotency class	3
isoclinism	isoclinic class of #332071
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	#25754	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

785 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)(9,10)(11,13)(12,14)(15,16)(17,18)(19,27)(20,25)(21,23)(22,26)(24,29)(28,31)(30,32);(1,3,4,8)(2,5,6,7)(10,13)(14,15)(18,23)(19,28)(20,22)(24,31)(27,32)(29,30);(10,14)(12,16)(13,15)(17,19,21,29)(18,24,23,27)(20,28,22,30)(25,32,26,31);(2,6)(3,8)(10,15)(12,16)(13,14)(17,20,21,22)(18,25,23,26)(19,28,29,30)(24,32,27,31)

Polycyclic presentation:

2^9|a^2=ehi,b^2=ehi,c^2=i,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,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

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 242797);