#6444407

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

2636 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fg,d^2=f,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

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, 6351603);