#8720377

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4124 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8627573);