#6465461

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

2514 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:

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

Polycyclic presentation:

2^9|c^2=gh,d^2=h,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,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, 6372657);