#374608

order	512
nilpotency class	3
isoclinism	isoclinic class of #196266
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

1423 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)(4,6)(5,8)(9,10)(11,19)(12,23,26,35)(13,17)(14,18)(15,21)(16,27,33,29)(20,37,24,40)(22,34)(25,32)(28,31)(30,39,36,38);(1,3,5,4)(2,6,8,7)(9,11,13,21)(10,15,17,19)(12,24,26,20)(14,28,25,22)(16,30,33,36)(18,34,32,31)(23,37,35,40)(27,38,29,39);(1,3,5,4)(2,7,8,6)(9,12,25,27,13,26,14,29)(10,16,32,23,17,33,18,35)(11,20,28,38,21,24,22,39)(15,30,34,40,19,36,31,37);(1,4,5,3)(2,7,8,6)(10,17)(15,19)(16,33)(18,32)(23,35)(30,36)(31,34)(37,40)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=efh,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,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, 281804);