#6575584

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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2252 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)(8,9)(11,12)(13,14,19,16)(15,25,26,21)(17,22,27,28)(18,23,20,24);(6,9)(10,12)(13,15,19,26)(14,21,16,25)(17,18,27,20)(22,24,28,23);(2,4)(13,16,19,14)(15,22,26,28)(17,25,27,21)(18,23,20,24);(5,7,8,11)(6,10,9,12)(13,17)(14,22)(15,18)(16,28)(19,27)(20,26)(21,24)(23,25);(13,18)(14,23)(15,26)(16,24)(19,20)(21,25)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,e^c=ei,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, 6482780);