#6444405

order	512
nilpotency class	3
isoclinism	isoclinic class of #25443
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂×D₄²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ 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) = 64

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

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

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,d^c=di

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, 6351601);