#7182014

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993899
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

1714 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7089210);