#734803

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

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

3858 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=gi,d^2=fi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 641999);