#6524722

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	#62824	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2916 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:

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

Polycyclic presentation:

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

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, 6431918);