#1260436

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×#547	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4718 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)(6,8)(9,10,12,14)(11,15,16,13)(18,23)(20,25)(24,26)(27,31);(11,16)(13,15)(17,18,22,23)(19,26,29,24)(20,21,25,30)(27,32,31,28);(1,2)(3,4)(5,6)(7,8)(9,11,12,16)(10,13,14,15)(17,19,22,29)(18,24,23,26)(20,27,25,31)(21,28,30,32);(5,6)(7,8)(17,20)(18,21)(19,27)(22,25)(23,30)(24,28)(26,32)(29,31);(1,2)(3,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,21)(18,25)(19,28)(20,23)(22,30)(24,31)(26,27)(29,32)

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^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, 1167632);