#8418539

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	#45038	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,22,21)(19,25,27,23)(20,28,30,24)(26,31,29,32)(33,34,38,41)(35,37,40,42)(36,39,47,45)(43,46,44,48);(2,6)(3,9)(8,13)(11,15)(17,18,22,21)(19,23,27,25)(20,28,30,24)(26,32,29,31)(33,35)(34,37)(36,44)(38,40)(39,48)(41,42)(43,47)(45,46);(4,12)(8,13)(11,15)(14,16)(18,21)(23,25)(24,28)(31,32)(35,40)(37,42)(43,44)(46,48);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19,22,27)(18,23,21,25)(20,29,30,26)(24,32,28,31)(33,34)(35,42)(36,45)(37,40)(38,41)(39,47)(43,46)(44,48);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,20)(18,24)(19,26)(21,28)(22,30)(23,31)(25,32)(27,29)(33,36)(34,39)(35,43)(37,46)(38,47)(40,44)(41,45)(42,48);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,21,22,18)(19,25,27,23)(20,28,30,24)(26,32,29,31)(33,37,38,42)(34,40,41,35)(36,46,47,48)(39,44,45,43)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8325735);