#94663

order	512
nilpotency class	6
isoclinism	isoclinic class of #94662
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1404
commutator	#382	C₂×C₄
Frattini, cosocle	#1961	C₂²
prime radical	C₂×#127	D₄
LCS	G ▷ #382 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1961 ◁ G
LECS	G ▷ #1961 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 193 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 40 points:

(1,2)(3,5)(4,7)(6,8)(9,10)(11,20)(12,24,25,35)(13,19,29,17)(14,18)(15,21,30,38)(16,26,34,28)(22,37,27,36)(23,33)(31,40,32,39);(2,5)(3,8)(6,7)(9,11,14,23)(10,15,18,32)(12,21,26,39)(13,27,29,22)(16,33,35,20)(17,30,19,31)(24,36,34,37)(25,40,28,38)

Polycyclic presentation:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1859);