#8186842

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	#46031	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4223 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)(17,18,22,26)(19,24,20,23)(21,25,29,32)(27,31,28,30)(33,34)(35,44)(36,39)(37,47)(38,42)(40,43)(41,46)(45,48);(1,3,6,9)(2,7,8,4)(5,11,14,16)(10,15,13,12)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,35,40,46)(34,41,43,44)(36,42,37,48)(38,39,45,47);(2,8)(3,9)(10,13)(11,16)(17,19,22,20)(18,23,26,24)(21,28,29,27)(25,31,32,30)(33,36,40,37)(34,39,43,47)(35,42,46,48)(38,41,45,44);(1,4,6,7)(2,9,8,3)(5,12,14,15)(10,16,13,11)(17,20,22,19)(18,24,26,23)(21,28,29,27)(25,31,32,30)(33,37,40,36)(34,39,43,47)(35,42,46,48)(38,44,45,41);(1,5,6,14)(2,10,8,13)(3,11,9,16)(4,12,7,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,38,40,45)(34,42,43,48)(35,39,46,47)(36,41,37,44);(1,2)(3,7)(4,9)(5,10)(6,8)(11,15)(12,16)(13,14)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,39)(34,36)(35,45)(37,43)(38,46)(40,47)(41,42)(44,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=gi,e^2=gh,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, 8094038);