#8979580

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
prime radical	#56339	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4276 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)(3,11)(4,9)(5,15)(6,7)(8,12)(10,16)(13,14)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(34,39)(35,41)(40,46)(42,47);(2,7)(4,14)(8,11)(10,15)(17,19,23,20)(18,24,22,25)(21,28,29,27)(26,32,30,31)(33,34)(35,36)(37,40)(38,39)(41,43)(42,44)(45,46)(47,48);(2,7)(3,12)(4,14)(5,16)(18,22)(19,20)(21,29)(31,32)(35,41)(36,43)(37,45)(40,46);(1,3)(2,8)(4,15)(5,13)(6,12)(7,11)(9,16)(10,14)(17,20,23,19)(18,25,22,24)(21,27,29,28)(26,31,30,32)(33,35,38,41)(34,36,39,43)(37,42,45,47)(40,44,46,48);(1,4,6,14)(2,9,7,13)(3,10,12,15)(5,11,16,8)(17,21,23,29)(18,26,22,30)(19,27,20,28)(24,31,25,32)(33,36)(34,35)(37,44)(38,43)(39,41)(40,42)(45,48)(46,47);(1,5)(2,10)(3,13)(4,11)(6,16)(7,15)(8,14)(9,12)(17,22,23,18)(19,25,20,24)(21,30,29,26)(27,32,28,31)(33,37,38,45)(34,40,39,46)(35,42,41,47)(36,44,43,48)

Polycyclic presentation:

2^9|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, 8886776);