#323227

order	512
nilpotency class	3
isoclinism	isoclinic class of #323197
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2541
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

650 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=egh,c^2=hi,d^2=ei,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dhi,f^c=fi,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230423);