#6465463

order	512
nilpotency class	3
isoclinism	isoclinic class of #6444406
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

2514 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:

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

Polycyclic presentation:

2^9|c^2=gh,d^2=hi,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,h^b=hi,e^c=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, 6372659);