#8963392

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	#42359	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3991 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=hi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8870588);