#720243

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×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

3592 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)(6,7)(11,15)(14,16)(17,18,22,25)(19,26,23,20)(21,24,30,32)(27,31,29,28);(10,13)(14,16)(17,19,22,23)(18,20,25,26)(21,29,30,27)(24,31,32,28);(5,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,27)(22,23)(24,28)(25,26)(29,30)(31,32);(1,2,3,4)(5,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,20)(18,23)(19,25)(21,28)(22,26)(24,29)(27,32)(30,31);(17,21,22,30)(18,24,25,32)(19,27,23,29)(20,28,26,31)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fg,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, 627439);