#696741

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#553	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4060 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)(11,14)(12,16)(13,15)(17,18)(19,24)(20,21)(22,23)(25,30)(26,32)(27,29)(28,31);(2,4)(6,8)(9,11,13,16)(10,14,15,12)(18,23)(21,25)(24,31)(27,32);(1,2,3,4)(5,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,24)(20,29)(21,27)(22,28)(23,31)(25,32)(26,30);(9,12)(10,11)(13,14)(15,16)(17,20)(18,25)(19,26)(21,23)(22,30)(24,27)(28,29)(31,32);(17,21,22,25)(18,20,23,30)(19,27,28,32)(24,29,31,26)

Polycyclic presentation:

2^9|b^2=fi,c^2=gh,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, 603937);