#7003545

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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2020 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)(5,6)(7,9)(8,10)(11,12)(13,14,19,22)(15,16,17,21)(18,28,26,23)(20,27,25,24);(5,6)(7,11)(8,10)(9,12)(13,15,18,24,19,17,26,27)(14,20,23,16,22,25,28,21);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,15)(17,22)(18,25)(19,21)(20,26)(23,24)(27,28);(7,12)(9,11)(13,17,19,15)(14,21,22,16)(18,24,26,27)(20,23,25,28);(6,10)(9,11)(14,22)(16,21)(20,25)(23,28)

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,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, 6910741);