#8506298

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

4131 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,e^2=ghi,f^2=hi,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, 8413494);