#8260871

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	#52249	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3930 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

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, 8168067);