#6529307

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	#62831	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2748 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=fi,c^2=h,d^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,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, 6436503);