#953312

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

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

1892 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ghi,c^2=fgh,d^2=hi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 860508);