#8549525

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

4121 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=gh,d^2=g,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8456721);