#8979558

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	#48421	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4084 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)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(17,18)(19,20)(21,29)(22,24)(23,25)(26,30)(27,31)(28,32);(1,3,7,4)(2,8,6,9)(5,12,13,11)(10,16,14,15)(17,19,22,25)(18,23,24,20)(21,27,30,32)(26,31,29,28)(33,34,38,39)(35,40,43,48)(36,41,46,45)(37,42,47,44);(2,6)(3,4)(5,13)(15,16)(18,24)(20,23)(21,30)(27,32)(34,39)(35,43)(36,46)(37,47);(1,4,7,3)(2,9,6,8)(5,11,13,12)(10,15,14,16)(17,20,22,23)(18,25,24,19)(21,28,30,31)(26,32,29,27)(33,35)(34,40)(36,42)(37,45)(38,43)(39,48)(41,47)(44,46);(1,5,7,13)(2,10,6,14)(3,11,4,12)(8,15,9,16)(17,18)(19,20)(21,26)(22,24)(23,25)(27,28)(29,30)(31,32)(33,36)(34,41)(35,44)(37,40)(38,46)(39,45)(42,43)(47,48);(1,6,7,2)(3,9,4,8)(5,14,13,10)(11,16,12,15)(17,21,22,30)(18,26,24,29)(19,27,25,32)(20,28,23,31)(33,37)(34,42)(35,45)(36,40)(38,47)(39,44)(41,43)(46,48)

Polycyclic presentation:

2^9|b^2=gi,d^2=h,e^2=gh,f^2=h,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, 8886754);