#1499597

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1622 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,c^2=fgi,d^2=ghi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef

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, 1406793);