#8934883

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	#2656	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4033 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)(3,5,9,15)(4,11,12,8)(10,16,13,14)(17,18)(19,23)(20,29)(21,26)(22,27)(24,28)(25,31)(30,32)(33,34)(35,41)(36,39)(37,45)(38,40)(42,46)(43,47)(44,48);(2,7)(3,9)(4,12)(14,16)(17,19,22,28)(18,23,27,24)(20,30,31,26)(21,29,32,25)(33,35,38,36)(34,39,40,41)(37,43,46,44)(42,47,45,48);(3,9)(5,15)(10,13)(14,16)(19,28)(21,32)(23,24)(26,30)(34,40)(37,46)(39,41)(43,44);(1,3,6,9)(2,5,7,15)(4,13,12,10)(8,16,11,14)(17,19)(18,24)(20,30)(21,29)(22,28)(23,27)(25,32)(26,31)(33,34)(35,41)(36,39)(37,42)(38,40)(43,48)(44,47)(45,46);(1,4,6,12)(2,8,7,11)(3,10,9,13)(5,14,15,16)(17,20,22,31)(18,25,27,29)(19,26,28,30)(21,23,32,24)(33,36,38,35)(34,41,40,39)(37,44,46,43)(42,48,45,47);(1,5)(2,9)(3,7)(4,14)(6,15)(8,13)(10,11)(12,16)(17,21)(18,26)(19,29)(20,23)(22,32)(24,31)(25,28)(27,30)(33,37,38,46)(34,42,40,45)(35,43,36,44)(39,47,41,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8842079);