#3856917

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1860 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)(3,7)(4,6)(5,8)(10,13)(11,15)(17,18,21,20)(19,23,24,22)(25,26,29,30)(27,31,34,38)(28,32,37,36)(33,39,35,40);(1,3,6,5)(2,7,4,8)(11,15)(14,16)(26,30)(31,38)(32,36)(39,40);(1,2,6,4)(3,8,5,7)(9,10)(11,14)(12,13)(15,16)(17,18,21,20)(19,22,24,23)(25,27,29,34)(26,31,30,38)(28,35,37,33)(32,40,36,39);(1,4,6,2)(3,7,5,8)(9,11)(10,14)(12,15)(13,16)(17,19,21,24)(18,22,20,23)(25,26)(27,31)(28,36)(29,30)(32,37)(33,40)(34,38)(35,39);(1,5,6,3)(2,7,4,8)(9,10)(11,14)(12,13)(15,16)(17,20,21,18)(19,23,24,22)(25,28,29,37)(26,32,30,36)(27,33,34,35)(31,39,38,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=ghi,d^2=fghi,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,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, 3764113);