#637528

order	512
nilpotency class	2
isoclinism	isoclinic class of #635984
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×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3786 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)(5,6,7,8)(17,18,22,23)(19,20,24,28)(21,29,30,25)(26,32,31,27);(9,10,12,14)(11,13,15,16)(18,23)(19,24)(25,29)(26,31);(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,20)(21,26)(22,24)(23,28)(25,27)(29,32)(30,31);(1,2)(3,4)(5,6)(7,8)(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,29)(30,32);(9,11,12,15)(10,13,14,16)(17,21,22,30)(18,25,23,29)(19,26,24,31)(20,27,28,32)

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,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, 544724);