#8406647

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

4194 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)(3,8)(4,14)(5,9)(6,10)(7,11)(12,16)(13,15)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32)(33,34)(35,43)(36,46)(37,47)(38,42)(39,44)(40,45)(41,48);(1,3,7,12)(2,8,11,16)(4,15,9,6)(5,10,14,13)(18,19)(23,25)(24,26)(30,31)(35,44)(37,48)(39,43)(41,47);(3,12)(6,15)(8,16)(10,13)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,35,38,44)(34,39,42,43)(36,37,45,48)(40,47,46,41);(1,4)(2,5)(3,6)(7,9)(8,13)(10,16)(11,14)(12,15)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,36)(34,40)(35,37)(38,45)(39,47)(41,43)(42,46)(44,48);(1,5)(2,9)(3,13)(4,11)(6,16)(7,14)(8,15)(10,12)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,37)(34,41)(35,45)(36,44)(38,48)(39,40)(42,47)(43,46);(1,6)(2,10)(3,4)(5,16)(7,15)(8,14)(9,12)(11,13)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,34)(35,43)(36,40)(37,41)(38,42)(39,44)(45,46)(47,48)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8313843);