#1265724

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3572 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1172920);