#1337781

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1820 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=f,e^2=hi,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, 1244977);