#720195

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

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

3232 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=f,d^2=hi,e^2=f,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, 627391);