#647619

order	512
nilpotency class	2
isoclinism	isoclinic class of #635984
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

1598 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

(3,7)(6,8)(9,10,14,16)(11,12,13,15)(25,26)(27,31)(28,36)(29,30)(32,37)(33,35)(34,39)(38,40);(1,2,5,4)(3,6,7,8)(9,11,14,13)(10,12,16,15)(17,18,21,24)(19,22,20,23)(26,30)(31,33)(32,36)(38,39);(9,12,14,15)(10,11,16,13)(17,18)(19,22)(20,23)(21,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,3)(2,6)(4,8)(5,7)(9,10)(11,12)(13,15)(14,16)(17,19,21,20)(18,22,24,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40);(1,4,5,2)(3,8,7,6)(9,13,14,11)(10,15,16,12)(17,20,21,19)(18,23,24,22)(25,28,29,37)(26,32,30,36)(27,34,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=g,b^2=fhi,c^2=g,d^2=fh,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^c=df

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, 554815);