#6576813

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573485
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 ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2825 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)(7,12)(8,10)(9,11)(13,14,19,20)(15,21,17,22)(16,26,23,18)(24,27,28,25);(5,7,9,8)(6,10,11,12)(13,15,19,17)(14,21,20,22)(16,27,23,25)(18,24,26,28);(2,4)(13,16)(14,18)(15,24)(17,28)(19,23)(20,26)(21,25)(22,27);(5,8,9,7)(6,10,11,12)(13,17,19,15)(14,22,20,21)(16,27,23,25)(18,24,26,28);(13,18,19,26)(14,23,20,16)(15,25,17,27)(21,28,22,24)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=ehi,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, 6484009);