#8359972

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

4095 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8267168);