#6437910

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437602
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4491 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345106);