#1126152

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

1994 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)(3,8)(4,6)(5,7)(10,14)(11,15)(17,18)(19,23)(20,24)(21,22)(25,26,30,29)(27,32,28,31);(1,3,5,4)(2,6,7,8)(9,10)(11,12)(13,14)(15,16)(17,19,21,24)(18,20,22,23)(25,27,30,28)(26,31,29,32);(3,4)(6,8)(10,14)(11,15)(18,22)(19,24)(25,26,30,29)(27,31,28,32);(1,4,5,3)(2,6,7,8)(9,11,13,15)(10,12,14,16)(17,18)(19,20)(21,22)(23,24)(25,28,30,27)(26,31,29,32);(1,2)(3,8)(4,6)(5,7)(9,12)(10,15)(11,14)(13,16)(17,20)(18,19)(21,23)(22,24)(25,29,30,26)(27,31,28,32)

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=fh,d^2=fg,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, 1033348);