#6444312

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439029
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
prime radical	#23399	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2641 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)(8,21)(9,13)(10,14)(11,16)(12,19)(15,22)(18,33)(20,31)(23,36)(24,29)(25,32)(26,30)(27,34)(28,35);(2,3)(5,7,10,19)(6,12,14,17)(8,22,20,11)(9,23,24,35)(13,28,29,36)(15,21,16,31)(18,25,34,26)(27,32,33,30);(1,2,4,3)(5,8,22,7,10,20,11,19)(6,12,16,31,14,17,15,21)(9,23,26,18,24,35,25,34)(13,27,32,28,29,33,30,36);(1,3,4,2)(5,9,10,24)(6,13,14,29)(7,18,19,34)(8,23,20,35)(11,25,22,26)(12,27,17,33)(15,30,16,32)(21,36,31,28);(7,19)(8,20)(12,17)(18,34)(21,31)(23,35)(27,33)(28,36)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fg,d^2=f,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,e^c=ei,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, 6351508);