#7143686

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994107
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2396 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

(1,2)(3,4)(7,11)(10,12)(13,14)(15,22)(16,17)(18,26)(19,21)(20,24)(23,28)(25,27);(6,9)(7,11)(13,15,16,23,19,25,18,24)(14,20,26,27,21,28,17,22);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,19,18)(14,17,21,26)(15,23,25,24)(20,22,28,27);(6,9)(10,12)(13,17,19,26)(14,16,21,18)(15,20,25,28)(22,24,27,23);(5,7,8,11)(6,10,9,12)(13,16,19,18)(14,17,21,26)(15,24,25,23)(20,27,28,22)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7050882);