#328442

order	512
nilpotency class	3
isoclinism	isoclinic class of #326735
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1013 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,23)(19,26,28,32)(20,25,22,30)(24,27,31,29);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(18,23)(20,22)(26,32)(27,29);(3,7)(6,8)(13,16)(17,19,21,28)(18,24,23,31)(20,29,22,27)(25,32,30,26);(1,3,4,7)(2,6,5,8)(17,20,21,22)(18,25,23,30)(19,27,28,29)(24,26,31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=e,f^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,d^c=de,f^c=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 235638);