#8621892

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	#46632	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4024 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8529088);