#100182

order	512
nilpotency class	3
isoclinism	isoclinic class of #100171
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

854 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 854 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 128

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=g,d^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cgi,e^b=eh,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, 7378);