#94678

order	512
nilpotency class	6
isoclinism	isoclinic class of #94678
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7416
commutator	#423	C₂×C₄
Frattini, cosocle	#1838	C₂²
prime radical	#3253	C₄
LCS	G ▷ #423 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1838 ◁ G
LECS	G ▷ #1838 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

133 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 133 conjugacy classes of subgroups — too many to draw. Only its 24 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1874);