#3301968

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
automorphism	Aut(G) order 1048576
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

1680 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)(3,6)(4,8)(5,7)(11,15)(14,16)(17,18,20,22)(19,23,24,21)(25,26,29,30)(27,33,34,31)(28,32,36,38)(35,40,37,39);(1,3,5,8)(2,6,7,4)(17,18,20,22)(19,21,24,23)(26,30)(27,34)(28,36)(39,40);(1,3)(2,4)(5,8)(6,7)(9,10,13,12)(11,14,15,16)(17,19)(18,21)(20,24)(22,23)(25,26)(27,33)(28,32)(29,30)(31,34)(35,40)(36,38)(37,39);(1,3)(2,4)(5,8)(6,7)(9,11,13,15)(10,14,12,16)(17,18)(19,21)(20,22)(23,24)(25,27)(26,31)(28,37)(29,34)(30,33)(32,40)(35,36)(38,39);(1,4)(2,3)(5,6)(7,8)(9,12,13,10)(11,16,15,14)(17,19)(18,21)(20,24)(22,23)(25,28)(26,32)(27,35)(29,36)(30,38)(31,39)(33,40)(34,37)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fg,c^2=gi,d^2=gi,e^2=gi,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, 3209164);