#6667615

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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

2813 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)(7,11)(8,10)(9,12)(13,14)(15,17)(16,27)(18,23)(19,21)(20,22)(24,25)(26,28);(5,7)(6,9)(8,12)(10,11)(13,14,18,23)(15,24,22,28)(16,21,19,27)(17,26,20,25);(2,4)(13,15,18,22)(14,20,23,17)(16,26,19,25)(21,28,27,24);(6,10)(9,11)(13,16,18,19)(14,21,23,27)(15,25,22,26)(17,28,20,24);(7,12)(9,11)(13,17,18,20)(14,22,23,15)(16,28,19,24)(21,26,27,25)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6574811);