#6573736

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573735
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

4508 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:

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

Polycyclic presentation:

2^9|d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,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

71 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72236799, #72237002, #72240982
maximal subgroup of	#72236799, #72236871, #72236873, …
maximal normal subgroup of	#72236799, #72236873, #72236997, …
maximal quotient of	#72251089, C₂×#6573736, #72262421, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480932);