#7149019

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993887
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1823 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7056215);