#8227414

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

4255 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)(33,34,40,44)(35,45,47,41)(36,38,37,42)(39,43,46,48);(2,7)(4,9)(8,14)(12,15)(17,18,22,23)(19,21,25,30)(20,27,29,24)(26,32,31,28)(33,35)(34,41)(36,43)(37,48)(38,39)(40,47)(42,46)(44,45);(5,13)(8,14)(10,16)(12,15)(18,23)(21,30)(24,27)(28,32)(33,36,40,37)(34,42,44,38)(35,43,47,48)(39,41,46,45);(1,3)(2,4)(5,10)(6,11)(7,9)(8,12)(13,16)(14,15)(17,19,22,25)(18,21,23,30)(20,26,29,31)(24,28,27,32)(33,37,40,36)(34,38,44,42)(35,43,47,48)(39,41,46,45);(1,4)(2,3)(5,12)(6,9)(7,11)(8,10)(13,15)(14,16)(17,20,22,29)(18,24,23,27)(19,26,25,31)(21,28,30,32)(33,38)(34,36)(35,46)(37,44)(39,47)(40,42)(41,48)(43,45);(1,5,6,13)(2,8,7,14)(3,10,11,16)(4,12,9,15)(17,21)(18,25)(19,23)(20,28)(22,30)(24,31)(26,27)(29,32)(33,39,40,46)(34,43,44,48)(35,38,47,42)(36,45,37,41)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=hi,e^2=i,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, 8134610);