#4861270

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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

1485 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fgh,c^2=fgh,d^2=ghi,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4768466);