#386075

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1605 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)(3,4)(9,10,15,13,16,20,14,19)(11,21,24,17,26,31,25,23)(12,27,28,18,30,33,29,34)(22,32,38,35,39,36,37,40);(6,8)(9,11)(10,17)(12,22)(13,31)(14,25)(15,24)(16,26)(18,32)(19,21)(20,23)(27,40)(28,38)(29,37)(30,39)(33,35)(34,36);(2,4)(5,6,7,8)(9,12,16,30)(10,18,20,34)(11,22,26,39)(13,27,19,33)(14,28,15,29)(17,32,23,36)(21,35,31,40)(24,37,25,38);(2,4)(9,13,14,20,16,19,15,10)(11,23,24,31,26,17,25,21)(12,27,28,34,30,33,29,18)(22,36,37,40,39,32,38,35)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293271);