#297208

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2819	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

727 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:

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

Polycyclic presentation:

2^9|a^2=ehi,c^2=fh,d^2=ehi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dhi,g^c=gh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 204404);