#290810

order	512
nilpotency class	3
isoclinism	isoclinic class of #267755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

873 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)(3,7)(4,6)(5,8)(9,10)(11,12)(13,16)(14,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,28)(29,32)(30,31);(1,3,4,8)(2,5,6,7)(9,11,14,13)(10,12,15,16)(17,19,22,29)(18,23,26,32)(20,30,25,27)(21,28,24,31);(9,12,11,10,14,16,13,15)(17,20,22,25)(18,24,26,21)(19,27)(23,31)(28,32)(29,30);(2,6)(5,7)(9,11,14,13)(10,12,15,16)(17,21,22,24)(18,25,26,20)(19,28,29,31)(23,27,32,30)

Polycyclic presentation:

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

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, 198006);