#6443289

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439025
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

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

4888 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=g,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^c=fi,h^c=hi,f^d=fi,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, 6350485);