#861528

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2401 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fh,c^2=fgi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

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, 768724);