#393372

order	512
nilpotency class	3
isoclinism	isoclinic class of #196092
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1476 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=efh,c^2=fi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^b=gi,e^d=ei,g^d=gi,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300568);