#836692

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

2022 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fg,c^2=f,e^2=fhi,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, 743888);