#364696

order	512
nilpotency class	3
isoclinism	isoclinic class of #196290
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1572 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=e,d^2=i,f^2=i,g^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, 271892);