#443560

order	512
nilpotency class	3
isoclinism	isoclinic class of #420406
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1931 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1931 conjugacy classes of subgroups — too many to draw. Only its 202 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 — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|b^2=e,c^2=eg,d^2=e,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,f^b=fi,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 350756);