#369243

order	512
nilpotency class	3
isoclinism	isoclinic class of #196299
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1310 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)(4,8)(5,6)(9,10,14,19,15,20,13,18)(11,16,22,31,23,33,29,32)(12,24,26,35,28,36,27,17)(21,37,38,30,39,34,25,40);(2,6)(7,8)(9,11,13,29,15,23,14,22)(10,16,18,32,20,33,19,31)(12,25,27,39,28,38,26,21)(17,34,36,30,35,37,24,40);(1,3,5,4)(2,7,6,8)(9,12)(10,17,20,35)(11,21,23,39)(13,26)(14,27)(15,28)(16,30)(18,24,19,36)(22,25,29,38)(31,37)(32,34)(33,40);(1,4,5,3)(2,7,6,8)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=efg,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,h^c=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, 276439);