#8630914

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
prime radical	#46696	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4024 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8538110);