#8979581

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

4227 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=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, 8886777);