#5277100

order	512
nilpotency class	2
isoclinism	isoclinic class of #5211343
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

1629 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=gh,c^2=fghi,d^2=fgh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,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, 5184296);