#6465462

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

Polycyclic presentation:

2^9|c^2=gh,d^2=h,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, 6372658);