#187668

order	512
nilpotency class	3
isoclinism	isoclinic class of #187667
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3561	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2270 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=i,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94864);