#1126172

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1996 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fi,c^2=h,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1033368);