#232531

order	512
nilpotency class	3
isoclinism	isoclinic class of #232528
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2196 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 2196 conjugacy classes of subgroups — too many to draw. Only its 198 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) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,g^b=gh,d^c=dh,e^c=ei,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139727);