#6443335

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437969
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4887 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)(7,17)(8,21)(9,14)(10,26)(11,16)(12,19)(13,24)(15,29)(18,35)(20,27)(22,34)(23,36)(25,32)(28,33)(30,31);(5,7,10,19,11,20,29,14)(6,12,15,17,16,9,26,27)(8,22,25,36,24,18,28,31)(13,30,32,35,21,23,33,34);(1,2,3,4)(5,8,11,24)(6,13,16,21)(7,18,20,22)(9,23,12,30)(10,28,29,25)(14,31,19,36)(15,33,26,32)(17,34,27,35);(5,9,11,12)(6,14,16,19)(7,15,20,26)(8,23,24,30)(10,17,29,27)(13,31,21,36)(18,33,22,32)(25,35,28,34);(8,24)(13,21)(18,22)(23,30)(25,28)(31,36)(32,33)(34,35)

Polycyclic presentation:

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

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, 6350531);