#3233668

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
automorphism	Aut(G) order 2097152
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

1588 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)(3,5,7,8)(9,10,12,14)(11,15,16,13)(18,22)(20,24)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,39)(38,40);(9,10,12,14)(11,13,16,15)(18,22)(19,23)(26,30)(28,37)(31,33)(34,38);(9,11)(10,13)(12,16)(14,15)(17,18,21,22)(19,20,23,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,37)(38,39);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,23)(18,20,22,24)(25,27,29,35)(26,31,30,33)(28,38,37,34)(32,40,36,39);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,16)(14,15)(17,20)(18,23)(19,22)(21,24)(25,28,29,37)(26,32,30,36)(27,34,35,38)(31,39,33,40)

Polycyclic presentation:

2^9|a^2=fi,b^2=fgh,c^2=fh,d^2=h,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef

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, 3140864);