#8479492

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	#39813	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4068 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)(4,13)(5,8)(6,9)(10,16)(11,14)(12,15)(17,18,22,27)(19,26,30,23)(20,24,25,21)(28,31,32,29)(36,44)(39,45)(43,47)(46,48);(1,3,6,12)(2,7,9,15)(4,11,5,10)(8,16,13,14)(17,19)(18,23)(20,28)(21,31)(22,30)(24,29)(25,32)(26,27)(33,34)(35,42)(36,45)(37,40)(38,41)(39,44)(43,46)(47,48);(1,4,6,5)(2,8,9,13)(3,10,12,11)(7,14,15,16)(17,20,22,25)(18,24,27,21)(19,28,30,32)(23,29,26,31)(33,35,38,40)(34,37,41,42)(36,43,44,47)(39,46,45,48);(1,5,6,4)(2,8,9,13)(3,11,12,10)(7,14,15,16)(17,18)(19,23)(20,24)(21,25)(22,27)(26,30)(28,29)(31,32)(33,34)(35,37)(36,39)(38,41)(40,42)(43,46)(44,45)(47,48);(1,3)(2,7)(4,10)(5,11)(6,12)(8,14)(9,15)(13,16)(17,21,22,24)(18,25,27,20)(19,29,30,31)(23,28,26,32)(33,36)(34,39)(35,43)(37,46)(38,44)(40,47)(41,45)(42,48);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,16)(12,15)(17,19)(18,26)(20,28)(21,29)(22,30)(23,27)(24,31)(25,32)(33,37)(34,40)(35,41)(36,46)(38,42)(39,47)(43,45)(44,48)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8386688);