#6533388

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×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3375 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)(13,14)(15,22)(16,21)(17,25)(18,20)(19,24)(23,27)(26,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,23)(14,19,20,28)(16,24,17,26)(21,27,25,22);(2,4)(6,9)(10,12)(14,20)(15,23)(21,25)(24,26);(5,7)(6,10)(8,11)(9,12)(13,14)(15,19)(16,25)(17,21)(18,20)(22,24)(23,28)(26,27);(7,11)(10,12)(13,16,18,17)(14,21,20,25)(15,24,23,26)(19,27,28,22)

Polycyclic presentation:

2^9|b^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,f^b=fi,d^c=di,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, 6440584);