#415573

order	512
nilpotency class	3
isoclinism	isoclinic class of #403352
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1239 conjugacy classes of subgroups; only the 248 normal subgroups are shown.

normal subgroups only

This group has 1239 conjugacy classes of subgroups — too many to draw. Only its 248 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322769);