#1262373

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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4114 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)(6,8)(11,15)(13,16)(18,23)(20,24)(26,31)(28,29);(9,10,12,14)(11,13,15,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,29)(28,32)(30,31);(1,2)(3,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,29,31,28);(5,6)(7,8)(9,11,12,15)(10,13,14,16)(17,20,22,24)(18,25,23,19)(21,29,30,28)(26,27,31,32);(1,2)(3,4)(5,6)(7,8)(17,21,22,30)(18,26,23,31)(19,27,25,32)(20,28,24,29)

Polycyclic presentation:

2^9|b^2=h,c^2=f,d^2=fh,e^2=f,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, 1169569);