#6533318

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

3307 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)(13,14)(15,24)(16,25)(17,22)(18,28)(19,23)(20,26)(21,27);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,21)(14,20,23,16)(17,26,18,25)(22,27,28,24);(6,9)(10,12)(13,16,19,20)(14,21,23,15)(17,27,18,24)(22,26,28,25);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,23,28)(15,25,21,26)(16,27,20,24);(7,11)(10,12)(13,17,19,18)(14,22,23,28)(15,25,21,26)(16,27,20,24)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6440514);