#696502

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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

1864 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)(3,7)(9,10,12,14)(11,13,15,16)(17,18)(19,20)(21,23)(22,24)(25,26)(27,34)(28,32)(29,33)(30,31)(35,40)(36,39)(37,38);(2,5)(6,8)(17,19,21,24)(18,20,23,22)(26,31)(27,32)(29,37)(36,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,23)(21,24)(25,27)(26,28)(29,35)(30,32)(31,34)(33,36)(37,39)(38,40);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,13,14,16)(17,20,21,22)(18,19,23,24)(25,28)(26,32)(27,31)(29,36)(30,34)(33,39)(35,38)(37,40);(25,29,30,37)(26,33,31,38)(27,35,32,39)(28,36,34,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=fhi,d^2=fg,e^2=f,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, 603698);