#362478

order	512
nilpotency class	3
isoclinism	isoclinic class of #220190
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

1244 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)(11,21,25,36,24,16,30,33)(12,26,28,35,27,17,29,34)(22,31,40,38,39,37,23,32);(1,3,5,4)(2,6,7,8)(9,11,15,30,13,24,14,25)(10,16,20,36,18,21,19,33)(12,23,29,22,27,40,28,39)(17,32,35,31,26,38,34,37);(2,7)(6,8)(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,31)(18,26)(19,35)(20,34)(21,37)(23,30)(24,39)(25,40)(32,36)(33,38);(1,4,5,3)(2,8,7,6)(9,12)(10,17)(11,23)(13,27)(14,28)(15,29)(16,32)(18,26)(19,34)(20,35)(21,38)(22,30)(24,40)(25,39)(31,36)(33,37)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,d^2=fi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,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, 269674);