#6533334

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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

3310 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)(16,24)(17,20)(18,21)(19,27)(23,26)(25,28);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,20,25)(14,16,17,26)(18,24,19,23)(21,28,27,22);(6,9)(10,12)(13,16)(14,15)(17,25)(18,22)(19,28)(20,26)(21,23)(24,27);(5,7)(6,10)(8,11)(9,12)(13,17,20,14)(15,16,25,26)(18,21,19,27)(22,23,28,24);(7,11)(10,12)(13,18,20,19)(14,21,17,27)(15,23,25,24)(16,22,26,28)

Polycyclic presentation:

2^9|b^2=i,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6440530);