#6607204

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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

2470 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)(15,17)(16,23)(18,21)(19,22)(20,26)(24,27)(25,28);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15,18,22,19,21,17,14)(16,24,26,20,27,23,25,28);(7,12)(9,10)(13,16,18,26,19,27,17,25)(14,20,15,23,22,28,21,24);(5,7)(6,10)(8,12)(9,11)(13,17,19,18)(14,15,22,21)(16,25,27,26)(20,23,28,24);(7,12)(9,10)(13,17,19,18)(14,15,22,21)(16,25,27,26)(20,23,28,24)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6514400);