#6529314

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500002
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2704 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(17,18)(19,23)(20,30,31,42)(21,26)(22,25)(24,32,39,34)(27,43,44,47)(28,38)(29,37)(33,41)(35,40)(36,45,48,46);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19,22,29)(18,23,25,37)(20,27,31,44)(21,33,35,28)(24,36,39,48)(26,41,40,38)(30,47,42,43)(32,46,34,45);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,20)(18,24)(19,27)(21,34)(22,31)(23,36)(25,39)(26,42)(28,46)(29,44)(30,40)(32,35)(33,45)(37,48)(38,47)(41,43);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,25)(23,37)(24,39)(26,40)(30,42)(36,48)(38,41)(43,47);(2,8)(6,9)(7,16)(14,15)(18,25)(23,37)(24,39)(26,40)(30,42)(36,48)(38,41)(43,47)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6436510);