#6529593

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500006
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	#60920	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2690 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:

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

Polycyclic presentation:

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

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, 6436789);