#291734

order	512
nilpotency class	3
isoclinism	isoclinic class of #269339
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

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

993 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)(3,7)(4,6)(5,8)(10,14)(11,12)(15,16)(17,18,23,26)(19,24,29,32)(20,21,25,22)(27,31,30,28);(1,3,4,8)(2,5,6,7)(10,15)(11,12)(14,16)(17,19,23,29)(18,24,26,32)(20,30,25,27)(21,28,22,31);(9,10,12,15,13,16,11,14)(17,20,22,18,23,25,21,26)(19,27,28,24,29,30,31,32);(2,6)(5,7)(10,16)(14,15)(17,21,23,22)(18,25,26,20)(19,28,29,31)(24,27,32,30)

Polycyclic presentation:

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

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, 198930);