#6529051

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499849
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	#60564	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2760 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,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, 6436247);