#361283

order	512
nilpotency class	3
isoclinism	isoclinic class of #222256
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1360 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)(3,6)(4,8)(5,7)(9,10,15,20,13,18,14,19)(11,21,26,33,24,16,25,34)(12,27,29,36,28,17,30,35)(22,31,23,32,39,37,40,38);(1,3,5,4)(2,6,7,8)(9,11,13,24)(10,16,18,21)(12,23,28,40)(14,26,15,25)(17,32,27,38)(19,34,20,33)(22,30,39,29)(31,36,37,35);(2,7)(6,8)(9,12)(10,17)(11,22)(13,28)(14,30)(15,29)(16,31)(18,27)(19,36)(20,35)(21,37)(23,26)(24,39)(25,40)(32,34)(33,38);(1,4,5,3)(2,8,7,6)(9,12)(10,17)(11,23)(13,28)(14,29)(15,30)(16,32)(18,27)(19,35)(20,36)(21,38)(22,26)(24,40)(25,39)(31,34)(33,37)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268479);