#6533560

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

3337 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)(15,24)(16,28)(17,21)(18,22)(19,23)(20,27)(25,26);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,19)(17,26)(18,27)(21,24)(22,28)(23,25);(6,9)(10,12)(13,16,19,15)(14,20,23,25)(17,27,18,26)(21,24,22,28);(5,7)(6,10)(8,11)(9,12)(13,14)(15,25)(16,20)(17,22)(18,21)(19,23)(24,26)(27,28);(7,11)(10,12)(13,17,19,18)(14,21,23,22)(15,26,16,27)(20,24,25,28)

Polycyclic presentation:

2^9|c^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,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, 6440756);