#721548

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	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=fi,e^2=fh,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, 628744);