#8417474

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	#42620	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3917 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)(3,8)(4,12)(5,10)(6,7)(9,13)(11,16)(14,15)(17,18)(19,24)(20,30)(21,25)(22,26)(23,27)(28,32)(29,31)(34,38)(36,43)(39,44)(42,47);(2,7)(4,13)(8,16)(10,14)(17,19)(18,24)(20,22)(21,31)(23,28)(25,29)(26,30)(27,32)(35,41)(39,44)(42,47)(46,48);(2,7)(5,15)(8,16)(9,12)(19,28)(22,29)(24,32)(26,31)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(1,3)(2,8)(4,14)(5,9)(6,11)(7,16)(10,13)(12,15)(17,20)(18,21)(19,29)(22,28)(23,25)(24,26)(27,30)(31,32)(33,35)(34,39)(36,42)(37,41)(38,44)(40,46)(43,47)(45,48);(1,4,6,13)(2,9,7,12)(3,10,11,14)(5,8,15,16)(17,21)(18,25)(19,26)(20,27)(22,24)(23,30)(28,31)(29,32)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48);(1,5,6,15)(2,10,7,14)(3,9,11,12)(4,8,13,16)(17,22,23,29)(18,26,27,31)(19,20,28,25)(21,24,30,32)

Polycyclic presentation:

2^9|c^2=gh,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, 8324670);