#9812990

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

4333 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9720186);