#8963705

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	#50115	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=ghi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8870901);