#6444303

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439025
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

2746 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fgi,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,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, 6351499);