#323247

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	#3169	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

677 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)(3,10)(4,6)(5,9)(7,12)(8,13)(11,14)(15,16)(17,18,22,24,20,25,21,26)(19,27,30,31,29,23,28,32);(2,6,9,8)(3,11)(7,10,15,14)(12,16)(17,19,21,28,20,29,22,30)(18,23,26,31,25,27,24,32);(1,2)(3,7)(4,8)(5,9)(6,13)(10,12)(11,15)(14,16)(18,24)(19,28)(21,22)(25,26)(29,30)(31,32);(1,3,4,12,5,11,13,16)(2,7,8,10,9,15,6,14)(18,25)(19,29)(24,26)(28,30)

Polycyclic presentation:

2^9|a^2=fi,b^2=egh,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, 230443);