#7142813

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993875
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

1911 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)(11,20)(12,17)(13,30)(14,18)(15,24)(16,28)(19,31)(21,33)(22,39)(23,34)(25,35)(26,32)(27,36)(29,37)(38,40);(2,5)(3,7)(9,11)(10,15)(12,26)(13,25)(14,23)(16,33)(17,38)(18,35)(19,34)(20,30)(21,29)(22,27)(24,31)(28,40)(32,36)(37,39);(1,2)(3,6)(4,5)(7,8)(9,12,14,29)(10,16,18,36)(11,21,23,26)(13,27,31,28)(15,32,35,33)(17,19,37,30)(20,39,34,38)(22,25,40,24);(2,5)(6,8)(9,12)(10,17)(11,22)(13,28)(14,29)(15,33)(16,30)(18,37)(19,36)(20,39)(21,24)(23,40)(25,26)(27,31)(32,35)(34,38);(1,3,4,7)(2,6,5,8)(10,18)(11,23)(16,36)(17,37)(19,30)(21,26)(22,40)(24,25)

Polycyclic presentation:

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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7050009);