#2228054

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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

1754 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=g,c^2=fi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

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, 2135250);