#6573729

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573729
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	D₄×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4601 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72236864, #72240991, #72240992
maximal subgroup of	#72236864, #72236873, #72236882, …
maximal normal subgroup of	#72236864, #72236873, #72237523, …
maximal quotient of	C₂×#6573729, #72251062, #72262411, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480925);