#382771

order	512
nilpotency class	3
isoclinism	isoclinic class of #190028
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1149 conjugacy classes of subgroups; only the 276 normal subgroups are shown.

normal subgroups only

This group has 1149 conjugacy classes of subgroups — too many to draw. Only its 276 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)(11,17)(12,20)(14,18)(16,22);(2,5)(6,8)(9,10)(11,18,17,14)(12,16)(13,15)(19,24,23,21)(20,22);(3,7)(6,8)(9,11,12,19,13,17,20,23)(10,14,16,21,15,18,22,24);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|b^2=f,c^2=h,d^2=e,h^2=i,b^a=be,c^a=cf,h^a=hi,c^b=cg,g^b=gi,d^c=de,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, 289967);