#6533532

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×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3392 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)(13,14,18,20)(15,22,24,28)(16,21,17,25)(19,26,27,23);(5,6)(7,10)(8,9)(11,12)(13,15,18,24)(14,19,20,27)(16,23,17,26)(21,28,25,22);(2,4)(6,9)(10,12)(13,14)(15,19)(16,25)(17,21)(18,20)(22,23)(24,27)(26,28);(5,7)(6,10)(8,11)(9,12)(14,20)(19,27)(21,25)(22,28);(7,11)(10,12)(13,16,18,17)(14,21,20,25)(15,23,24,26)(19,28,27,22)

Polycyclic presentation:

2^9|a^2=i,b^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,f^b=fi,d^c=di,e^c=ei,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, 6440728);