#8202392

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

4257 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=h,f^2=g,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, 8109588);