#6683529

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2344 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)(6,9)(7,11)(13,14)(15,20)(16,21)(17,25)(18,22)(19,23)(24,26)(27,28);(5,6)(7,10)(8,9)(11,12)(13,15,19,25)(14,17,23,20)(16,21,27,28)(18,22,24,26);(1,2,3,4)(13,16)(14,21)(15,24)(17,22)(18,25)(19,27)(20,26)(23,28);(5,7,8,11)(6,10,9,12)(13,17,19,20)(14,15,23,25)(16,26,27,22)(18,28,24,21);(6,9)(10,12)(13,18,19,24)(14,22,23,26)(15,16,25,27)(17,28,20,21)

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6590725);