#7143667

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993979
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

2324 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)(13,14,19,22)(15,23,26,28)(16,21,18,17)(20,27,25,24);(6,9)(7,11)(13,15,16,27,19,26,18,24)(14,20,17,28,22,25,21,23);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,19,18)(14,17,22,21)(15,24,26,27)(20,23,25,28);(6,9)(10,12)(13,17)(14,16)(15,25)(18,22)(19,21)(20,26)(23,24)(27,28);(5,7,8,11)(6,10,9,12)(13,16,19,18)(14,21,22,17)(15,24,26,27)(20,28,25,23)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7050863);