#7149351

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2013 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,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,e^c=ei,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, 7056547);