#8720619

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	#56307	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4036 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gi,d^2=i,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, 8627815);