#248722

order	512
nilpotency class	3
isoclinism	isoclinic class of #248647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1218 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 155918);