#100190

order	512
nilpotency class	3
isoclinism	isoclinic class of #9495
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁶	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁

1106 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 1106 conjugacy classes of subgroups — too many to draw. Only its 106 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 — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=gh,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dg,e^b=eh,f^b=fg,d^c=dh,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7386);