#3827179

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1368 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fg,c^2=fi,d^2=fgi,e^2=fgi,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, 3734375);