#6489662

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470122
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
prime radical	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2987 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2)(3,4)(5,6)(7,11)(8,12)(9,13)(10,14)(15,18)(16,19)(17,20)(21,22)(23,32)(24,30)(25,29)(26,31)(27,34)(28,33)(35,36);(2,4)(5,7,10,17)(6,11,14,20)(8,16,9,15)(12,19,13,18)(21,23)(22,27)(24,33)(25,34)(26,35)(28,31)(29,36)(30,32);(5,8,10,9)(6,12,14,13)(7,15,17,16)(11,18,20,19)(21,23)(22,28)(24,33)(25,34)(26,35)(27,31)(29,32)(30,36);(5,9,10,8)(6,13,14,12)(7,16,17,15)(11,19,20,18)(21,24,26,25)(22,29,31,30)(23,33,35,34)(27,36,28,32);(6,14)(11,20)(12,13)(18,19)(21,24,26,25)(22,30,31,29)(23,33,35,34)(27,32,28,36)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6396858);