#8841413

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

4270 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)(3,10)(4,5)(6,7)(8,11)(9,15)(12,13)(14,16)(17,18,22,25)(19,20,23,27)(21,24,30,32)(26,28,29,31)(33,34,38,42)(35,43,46,39)(36,47,45,40)(37,48,44,41);(2,7)(5,9)(8,10)(13,16)(17,19)(18,20)(21,29)(22,23)(24,31)(25,27)(26,30)(28,32)(33,35,38,46)(34,39,42,43)(36,44,45,37)(40,41,47,48);(3,11)(8,10)(12,14)(13,16)(19,23)(20,27)(26,29)(28,31)(33,36,38,45)(34,40,42,47)(35,44,46,37)(39,41,43,48);(1,3,6,11)(2,8,7,10)(4,14,15,12)(5,13,9,16)(17,20,22,27)(18,23,25,19)(21,28,30,31)(24,29,32,26)(33,36)(34,40)(35,44)(37,46)(38,45)(39,41)(42,47)(43,48);(1,4)(2,9)(3,12)(5,7)(6,15)(8,13)(10,16)(11,14)(17,18)(19,20)(21,24)(22,25)(23,27)(26,28)(29,31)(30,32)(33,37,38,44)(34,41,42,48)(35,45,46,36)(39,40,43,47);(1,5,6,9)(2,4,7,15)(3,13,11,16)(8,14,10,12)(17,21)(18,24)(19,26)(20,28)(22,30)(23,29)(25,32)(27,31)(33,34)(35,43)(36,40)(37,41)(38,42)(39,46)(44,48)(45,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

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, 8748609);