#9783021

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

4098 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)(3,9)(4,6)(5,8)(7,11)(10,16)(12,14)(13,15)(17,18,22,25)(19,23,28,31)(20,21,29,24)(26,27,32,30)(33,34)(35,38)(36,37)(39,42)(40,46)(41,47)(43,48)(44,45);(1,3)(2,7)(4,9)(5,10)(6,11)(8,15)(12,16)(13,14)(17,19,22,28)(18,23,25,31)(20,26,29,32)(21,27,24,30)(33,35,39,41)(34,38,42,47)(36,44,46,43)(37,45,40,48);(2,4)(7,9)(8,12)(15,16)(17,20,22,29)(18,24,25,21)(19,26,28,32)(23,30,31,27)(33,36,39,46)(34,40,42,37)(35,43,41,44)(38,45,47,48);(1,4,6,2)(3,9,11,7)(5,12,14,8)(10,16,13,15)(17,20)(18,21)(19,26)(22,29)(23,27)(24,25)(28,32)(30,31)(33,36)(34,37)(35,44)(38,45)(39,46)(40,42)(41,43)(47,48);(1,5,6,14)(2,8,4,12)(3,10,11,13)(7,15,9,16)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(33,37)(34,36)(35,45)(38,44)(39,40)(41,48)(42,46)(43,47);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,14)(12,15)(17,21)(18,20)(19,27)(22,24)(23,26)(25,29)(28,30)(31,32)(33,38)(34,41)(35,42)(36,45)(37,43)(39,47)(40,44)(46,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=g,e^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, 9690217);