#6676718

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2022 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)(7,10)(8,12)(9,11)(13,14)(15,19)(16,20)(17,22)(18,21)(23,27)(24,25)(26,28);(5,7,10,6)(8,12,9,11)(13,15,18,26)(14,19,21,28)(16,24,22,27)(17,23,20,25);(2,4)(15,23)(16,22)(17,20)(19,27)(24,28)(25,26);(5,8,10,9)(6,11,7,12)(13,16,18,22)(14,20,21,17)(15,24,26,27)(19,25,28,23);(5,9,10,8)(6,11,7,12)(14,21)(15,25)(17,20)(19,27)(23,26)(24,28)

Polycyclic presentation:

2^9|b^2=fi,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6583914);