#7148117

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,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, 7055313);