#8139604

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	#45726	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4223 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)(3,10,4,7)(5,13,14,8)(11,15,12,16)(17,18)(19,26)(20,24)(21,31)(22,32)(23,25)(27,28)(29,30)(33,34,38,42)(35,41,37,39)(36,44,46,40)(43,47,45,48);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(19,20)(24,26)(27,29)(28,30)(33,35,38,37)(34,39,42,41)(36,45,46,43)(40,48,44,47);(17,19,23,20)(18,24,25,26)(21,27,32,29)(22,28,31,30)(33,35,38,37)(34,39,42,41)(36,43,46,45)(40,47,44,48);(1,2)(3,7)(4,10)(5,8)(6,9)(11,15)(12,16)(13,14)(17,20,23,19)(18,24,25,26)(21,27,32,29)(22,30,31,28)(33,36)(34,40)(35,43)(37,45)(38,46)(39,47)(41,48)(42,44);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,21)(18,22)(19,27)(20,29)(23,32)(24,28)(25,31)(26,30)(33,37,38,35)(34,41,42,39)(36,43,46,45)(40,47,44,48);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,22)(18,21)(19,28)(20,30)(23,31)(24,27)(25,32)(26,29)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,46)(45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8046800);