#8418441

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	#53063	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4003 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=g,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, 8325637);