#9912856

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4155 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9820052);