#9016550

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#51924	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3920 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=ghi,d^2=hi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8923746);