#8963378

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

4231 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=hi,e^2=hi,f^2=gh,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, 8870574);