#6438868

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437608
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

3108 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:

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

Polycyclic presentation:

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

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, 6346064);