#6583377

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2249 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)(8,9)(11,12)(13,14,18,19)(15,23,25,20)(16,21,26,28)(17,22,24,27);(6,9)(10,12)(14,19)(15,20)(16,26)(17,22)(23,25)(24,27);(1,2,3,4)(13,15,18,25)(14,20,19,23)(16,27,26,22)(17,21,24,28);(5,7,8,11)(6,10,9,12)(13,16,18,26)(14,21,19,28)(15,24,25,17)(20,22,23,27);(13,17,18,24)(14,22,19,27)(15,21,25,28)(16,23,26,20)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6490573);