#6442613

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437478
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62859	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4490 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,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, 6349809);