#257026

order	512
nilpotency class	3
isoclinism	isoclinic class of #245999
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	#2847	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1061 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)(4,6)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,20)(21,25)(22,26)(23,27)(24,30)(28,32)(29,31);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19,23,29)(18,20,27,31)(21,24,22,28)(25,30,26,32);(1,4,5,3)(2,6,7,8)(13,15)(17,20)(18,24)(19,26)(21,30)(22,32)(23,31)(25,29)(27,28);(2,7)(6,8)(10,14)(13,15)(17,21,23,22)(18,25,27,26)(19,24,29,28)(20,30,31,32)

Polycyclic presentation:

2^9|b^2=eh,c^2=ei,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,f^c=fh,g^c=gh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 164222);