#8339959

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

4226 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)(3,11)(4,14)(5,10)(6,7)(8,12)(9,15)(13,16)(18,19)(23,25)(24,26)(30,31)(33,34,38,41)(35,42,36,39)(37,40,45,48)(43,47,44,46);(2,7)(3,12)(9,14)(10,13)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(35,36)(39,42)(43,44)(46,47);(2,7)(5,16)(8,11)(9,14)(21,29)(24,26)(27,32)(30,31)(33,35,38,36)(34,39,41,42)(37,44,45,43)(40,47,48,46);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,30,32,31)(33,36,38,35)(34,39,41,42)(37,44,45,43)(40,46,48,47);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,34)(35,39)(36,42)(37,40)(38,41)(43,46)(44,47)(45,48);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,48)(42,47)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=i,d^2=gi,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8247155);