#8720588

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	#46699	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4050 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)(3,7)(4,12)(5,9)(6,10)(8,13)(11,14)(15,16)(17,18,22,26)(19,21,25,23)(20,28,30,24)(27,29,32,31)(33,34)(35,42)(36,39)(37,44)(38,41)(40,46)(43,45)(47,48);(3,10)(6,7)(11,16)(14,15)(17,19,22,25)(18,23,26,21)(20,27,30,32)(24,29,28,31)(34,36)(35,41)(40,44)(43,47);(1,3,5,10)(2,6,9,7)(4,11,13,16)(8,14,12,15)(17,19,22,25)(18,23,26,21)(20,27,30,32)(24,29,28,31)(33,35)(34,38)(36,42)(37,43)(39,41)(40,45)(44,48)(46,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,29)(28,31)(30,32)(33,36,39,34)(35,42,41,38)(37,44,46,40)(43,48,47,45);(1,4,5,13)(2,8,9,12)(3,11,10,16)(6,14,7,15)(17,20,22,30)(18,24,26,28)(19,27,25,32)(21,31,23,29)(33,37,39,46)(34,40,36,44)(35,43,41,47)(38,45,42,48);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,27,28,32)(33,38,39,42)(34,41,36,35)(37,45,46,48)(40,47,44,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gi,d^2=ghi,e^2=h,f^2=g,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, 8627784);