#1994641

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
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

1437 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=gh,c^2=ghi,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1901837);