#327300

order	512
nilpotency class	3
isoclinism	isoclinic class of #326783
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#14811	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1396 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 1396 conjugacy classes of subgroups — too many to draw. Only its 156 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)(3,6)(4,5)(7,8)(9,10,12,15)(11,13,16,14)(17,18,20,22)(19,24,21,23);(2,5)(6,8)(18,22)(19,21);(1,3)(2,6)(4,7)(5,8)(10,13)(11,16)(14,15)(18,23)(19,21)(22,24);(3,7)(6,8)(17,19,20,21)(18,24,22,23)

Polycyclic presentation:

2^9|a^2=h,d^2=i,f^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=di,f^b=fi,d^c=de,f^c=fh

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, 234496);