#7148116

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

1675 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,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, 7055312);