#6573672

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573671
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4549 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:

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

Polycyclic presentation:

2^9|c^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480868);