#749053

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

1826 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fgh,c^2=f,d^2=g,e^2=hi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,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, 656249);