#8814020

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	#49842	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4140 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=ghi,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8721216);