#6607187

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2403 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6514383);