#8746322

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4128 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)(5,10)(6,11)(8,12)(13,15)(14,16)(17,18,23,25)(19,27,21,24)(20,26,31,22)(28,32,30,29)(33,34,38,42)(35,39,45,48)(36,46,44,40)(37,41,43,47);(1,3)(2,7)(4,13)(5,14)(6,12)(8,11)(9,15)(10,16)(17,19,23,21)(18,24,25,27)(20,30,31,28)(22,29,26,32)(33,35,38,45)(34,39,42,48)(36,37,44,43)(40,47,46,41);(3,12)(7,8)(13,16)(14,15)(18,25)(20,31)(24,27)(28,30)(33,36,38,44)(34,40,42,46)(35,43,45,37)(39,41,48,47);(1,3)(2,8)(4,13)(5,15)(6,12)(7,11)(9,14)(10,16)(17,20,23,31)(18,26,25,22)(19,28,21,30)(24,29,27,32)(33,36)(34,40)(35,43)(37,45)(38,44)(39,41)(42,46)(47,48);(1,4)(2,9)(3,13)(5,11)(6,10)(7,15)(8,14)(12,16)(17,21,23,19)(18,27,25,24)(20,30,31,28)(22,29,26,32)(33,34)(35,39)(36,40)(37,47)(38,42)(41,43)(44,46)(45,48);(1,5,6,9)(2,10,11,4)(3,14,12,15)(7,16,8,13)(17,22,23,26)(18,20,25,31)(19,29,21,32)(24,30,27,28)(33,37,38,43)(34,41,42,47)(35,44,45,36)(39,40,48,46)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8653518);