#7142823

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

1919 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)(11,21)(12,25)(13,24)(14,33)(15,19)(16,18)(17,29)(20,34)(22,31)(23,32)(26,27)(28,36)(30,37)(35,40)(38,39);(2,5)(3,7)(9,11)(10,16)(12,26)(13,19)(14,24)(15,23)(17,31)(18,34)(20,32)(21,33)(22,30)(25,40)(27,36)(28,38)(29,35)(37,39);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,22)(13,31)(14,28)(15,30)(16,27)(18,38)(19,36)(20,37)(21,39)(23,26)(24,35)(25,33)(29,34)(32,40);(2,5)(6,8)(9,13)(10,18)(11,20)(12,27)(14,21)(15,16)(17,35)(19,24)(22,25)(23,33)(26,37)(28,40)(29,39)(30,31)(32,34)(36,38);(1,3,4,7)(2,6,5,8)(10,19)(11,23)(17,36)(18,24)(20,33)(22,26)(25,37)(35,38)

Polycyclic presentation:

2^9|e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,g^a=gi,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, 7050019);