#323163

order	512
nilpotency class	3
isoclinism	isoclinic class of #323133
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

651 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)(3,8)(4,12)(5,7)(6,10)(9,15)(11,14)(13,16)(17,18,22,27,23,28,29,26)(19,20,21,24,30,25,32,31);(2,7,10,9)(3,11,13,12)(17,19,29,32,23,30,22,21)(18,24,27,25,28,31,26,20);(1,3,6,13)(2,8,10,16)(4,9,14,7)(5,12,15,11)(17,20)(18,19)(21,26)(22,31)(23,25)(24,29)(27,32)(28,30);(1,4,5,8,6,14,15,16)(2,3,9,12,10,13,7,11)(17,21,29,19,23,32,22,30)(18,25,27,31,28,20,26,24)

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=di,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, 230359);