#6432003

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431795
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3269 conjugacy classes of subgroups; only the 986 normal subgroups are shown.

normal subgroups only

This group has 3269 conjugacy classes of subgroups — too many to draw. Only its 986 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=f,d^2=i,e^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ch,g^b=gi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339199);