#9970064

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

4053 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)(3,7)(4,9)(5,10)(6,11)(8,13)(12,15)(14,16)(17,18)(19,22)(20,27)(21,24)(23,28)(25,31)(26,30)(29,32)(33,34,38,43)(35,41,46,39)(36,40,42,37)(44,45,48,47);(1,3)(2,7)(4,10)(5,9)(6,13)(8,11)(12,14)(15,16)(17,19,21,26)(18,22,24,30)(20,25,28,32)(23,29,27,31)(33,35)(34,39)(36,44)(37,45)(38,46)(40,47)(41,43)(42,48);(1,3,6,13)(2,7,11,8)(4,12,14,10)(5,9,15,16)(17,18,21,24)(19,22,26,30)(20,23,28,27)(25,29,32,31)(33,35)(34,39)(36,44)(37,45)(38,46)(40,47)(41,43)(42,48);(1,3)(2,8)(4,12)(5,9)(6,13)(7,11)(10,14)(15,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,36,38,42)(34,40,43,37)(35,44,46,48)(39,47,41,45);(1,4)(2,9)(3,12)(5,8)(6,14)(7,15)(10,13)(11,16)(17,19)(18,22)(20,25)(21,26)(23,29)(24,30)(27,31)(28,32)(33,35)(34,41)(36,44)(37,47)(38,46)(39,43)(40,45)(42,48);(1,5)(2,10)(3,9)(4,7)(6,15)(8,14)(11,12)(13,16)(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,32)(30,31)(33,37,38,40)(34,42,43,36)(35,45,46,47)(39,48,41,44)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ghi,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dghi,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, 9877260);