#369295

order	512
nilpotency class	3
isoclinism	isoclinic class of #196350
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, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1288 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)(9,10,13,18,14,17,25,31)(11,15,21,28,20,27,35,39)(12,16,23,30,22,29,24,32)(19,26,34,37,33,38,36,40);(2,6)(7,8)(9,11,13,21,14,20,25,35)(10,15,18,28,17,27,31,39)(12,19,23,34,22,33,24,36)(16,26,30,37,29,38,32,40);(1,3,5,4)(2,7,6,8)(9,12)(10,16,17,29)(11,19,20,33)(13,24)(14,22)(15,26)(18,32,31,30)(21,36,35,34)(23,25)(27,38)(28,40)(37,39);(1,4,5,3)(2,7,6,8)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=efg,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,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, 276491);