#4308737

order	512
nilpotency class	2
isoclinism	isoclinic class of #4307937
automorphism	Aut(G) order 12582912
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

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

1745 conjugacy classes of subgroups; only the 538 normal subgroups are shown.

normal subgroups only

This group has 1745 conjugacy classes of subgroups — too many to draw. Only its 538 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=f,d^2=h,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4215933);