#8271894

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

4246 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=i,d^2=i,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8179090);