#6465515

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442590
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2502 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)(7,17)(8,21)(9,14)(10,15)(11,16)(12,19)(13,23)(18,32)(20,27)(22,30)(24,29)(25,28)(26,35)(31,34)(33,36);(2,3)(5,7)(6,12)(8,18)(9,19)(10,25)(11,20)(13,26)(14,27)(15,17)(16,28)(21,31)(22,33)(23,34)(24,35)(29,36)(30,32);(1,2,4,3)(5,8,10,22,11,24,9,23)(6,13,14,29,16,30,15,21)(7,18,25,33,20,35,19,34)(12,26,27,36,28,32,17,31);(1,3,4,2)(5,6)(7,12)(8,13)(9,15)(10,14)(11,16)(17,19)(18,26)(20,28)(21,23)(22,29)(24,30)(25,27)(31,34)(32,35)(33,36);(5,9,11,10)(6,14,16,15)(7,19,20,25)(8,22,24,23)(12,27,28,17)(13,21,30,29)(18,33,35,34)(26,31,32,36)

Polycyclic presentation:

2^9|c^2=ghi,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=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, 6372711);