#814017

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
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

2051 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fgh,c^2=fghi,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 721213);