#8188202

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4067 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)(4,9)(5,13)(6,8)(10,14)(11,12)(15,16)(18,23)(19,20)(26,30)(27,28)(33,34)(35,36)(37,46)(38,41)(39,42)(40,43)(44,47)(45,48);(1,3,6,9)(2,7,8,4)(5,11,14,16)(10,15,13,12)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,35)(34,37)(36,43)(38,44)(39,45)(40,46)(41,42)(47,48);(2,8)(3,9)(10,13)(11,16)(17,19,22,20)(18,24,23,25)(21,28,29,27)(26,32,30,31)(33,36,40,37)(34,35,43,46)(38,47,45,42)(39,41,44,48);(1,4,6,7)(2,9,8,3)(5,12,14,15)(10,16,13,11)(17,20,22,19)(18,25,23,24)(21,28,29,27)(26,32,30,31)(33,37,40,36)(34,35,43,46)(38,42,45,47)(39,41,44,48);(1,5,6,14)(2,10,8,13)(3,11,9,16)(4,12,7,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,38,40,45)(34,41,43,48)(35,44,46,39)(36,47,37,42);(1,2)(3,7)(4,9)(5,10)(6,8)(11,15)(12,16)(13,14)(17,21,22,29)(18,26,23,30)(19,27,20,28)(24,31,25,32)(33,39,40,44)(34,42,43,47)(35,45,46,38)(36,48,37,41)

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095398);