#94710

order	512
nilpotency class	6
isoclinism	isoclinic class of #94670
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1402
commutator	#384	C₂×C₄
Frattini, cosocle	#1957	C₂²
prime radical	C₂×#127	D₄
LCS	G ▷ #384 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1957 ◁ G
LECS	G ▷ #1957 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

180 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 26 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 — 2 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fg,e^2=hi,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fh,g^a=gh,h^a=hi,c^b=cf,d^b=defghi,f^b=fh,g^b=gh,d^c=dgh,e^c=ehi,f^c=fi,g^c=gi,e^d=ehi,f^d=fi,g^d=gi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1906);