#6444413

order	512
nilpotency class	3
isoclinism	isoclinic class of #6444406
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

2654 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:

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

Polycyclic presentation:

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

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, 6351609);