#7148115

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

1691 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)(9,10,14,19,15,11,20,18)(12,23,26,21,27,36,25,16)(13,22,31,28,32,17,30,35)(24,38,39,33,29,37,40,34);(2,5)(3,7)(9,11)(10,15)(12,16)(13,28)(14,18)(17,31)(19,20)(21,27)(22,30)(23,26)(24,38)(25,36)(29,37)(32,35)(33,39)(34,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,21)(13,29)(14,25)(15,27)(17,34)(18,23)(19,36)(20,26)(22,33)(24,32)(28,38)(30,40)(31,39)(35,37);(2,5)(6,8)(9,13)(10,17)(11,22)(12,24)(14,31)(15,32)(16,33)(18,28)(19,35)(20,30)(21,34)(23,37)(25,40)(26,39)(27,29)(36,38);(1,3,4,7)(2,6,5,8)(10,11)(16,21)(17,22)(18,19)(23,36)(28,35)(33,34)(37,38)

Polycyclic presentation:

2^9|a^2=g,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, 7055311);