#6432002

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3633 conjugacy classes of subgroups; only the 986 normal subgroups are shown.

normal subgroups only

This group has 3633 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) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 24 points:

(1,2,5,7)(3,8,4,6)(9,10)(11,14)(12,15)(13,16)(17,21)(18,20)(19,22)(23,24);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,17)(13,18)(15,21)(16,20)(19,23)(22,24);(1,4,5,3)(2,6,7,8)(14,20)(21,24);(9,12)(10,15)(11,17)(13,19)(14,21)(16,22)(18,23)(20,24);(12,19)(15,22)(17,23)(21,24)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=f,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, 6339198);