#8479513

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

4156 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=i,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=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, 8386709);