#8143155

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3909 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)(4,10)(5,6)(7,9)(8,12)(13,15)(14,16)(17,18,22,23)(19,26,29,24)(20,21,30,25)(27,31,32,28)(33,34)(35,43)(36,41)(37,42)(38,39)(40,46)(44,48)(45,47);(1,3,8,7)(2,9,12,11)(4,13,5,14)(6,15,10,16)(18,23)(20,30)(24,26)(27,32)(34,39)(40,43)(41,42)(47,48);(1,4,8,5)(2,10,12,6)(3,13,7,14)(9,16,11,15)(17,19)(18,24)(20,27)(21,28)(22,29)(23,26)(25,31)(30,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,46)(44,47)(45,48);(1,5,8,4)(2,10,12,6)(3,14,7,13)(9,16,11,15)(17,20,22,30)(18,25,23,21)(19,27,29,32)(24,31,26,28)(33,35)(34,40)(36,44)(37,45)(38,46)(39,43)(41,47)(42,48);(1,6,8,10)(2,5,12,4)(3,15,7,16)(9,14,11,13)(17,21,22,25)(18,20,23,30)(19,28,29,31)(24,27,26,32)(33,36,38,37)(34,41,39,42)(35,44,46,45)(40,47,43,48);(1,7,8,3)(2,11,12,9)(4,14,5,13)(6,16,10,15)(17,19)(18,24)(20,27)(21,28)(22,29)(23,26)(25,31)(30,32)(33,37,38,36)(34,42,39,41)(35,45,46,44)(40,48,43,47)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=ghi,d^2=gi,e^2=g,f^2=gh,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, 8050351);