#6467082

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
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 ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2078 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:

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

Polycyclic presentation:

2^9|c^2=h,d^2=hi,e^2=g,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374278);