#6443352

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437995
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	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

5158 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)(7,11)(8,13)(9,22)(10,15)(12,21)(14,23)(16,25)(17,31)(18,27)(19,24)(20,28)(26,32)(29,34)(30,35)(33,36);(5,7)(6,11)(8,19,21,16)(9,17)(10,18)(12,24,13,25)(14,26)(15,27)(20,33,34,30)(22,31)(23,32)(28,35,29,36);(1,2,3,4)(5,8)(6,12)(7,16)(9,20)(10,21)(11,24)(13,15)(14,28)(17,30)(18,19)(22,29)(23,34)(25,27)(26,35)(31,36)(32,33);(5,8)(6,13)(7,16)(9,20)(10,21)(11,25)(12,15)(14,29)(17,30)(18,19)(22,28)(23,34)(24,27)(26,36)(31,35)(32,33);(5,9,10,23)(6,14,15,22)(7,17,18,32)(8,20,21,34)(11,26,27,31)(12,28,13,29)(16,30,19,33)(24,35,25,36)

Polycyclic presentation:

2^9|b^2=f,c^2=gi,e^2=i,b^a=bf,c^a=cg,e^a=ei,h^a=hi,c^b=ch,d^b=dh,h^b=hi,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, 6350548);