#8334539

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	#56836	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3905 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)(3,11,12,8)(4,13,15,9)(5,10,16,14)(17,18,23,20)(19,22,25,26)(21,24,30,28)(27,29,31,32)(35,43)(36,45)(39,46)(40,48);(2,7)(3,12)(9,13)(10,14)(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,34,38,42)(35,39,43,46)(36,40,45,48)(37,41,44,47);(2,7)(5,16)(8,11)(9,13)(19,25)(22,26)(27,31)(29,32)(33,35)(34,39)(36,44)(37,45)(38,43)(40,47)(41,48)(42,46);(1,3)(2,8)(4,14)(5,9)(6,12)(7,11)(10,15)(13,16)(17,20,23,18)(19,26,25,22)(21,24,30,28)(27,29,31,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,4,6,15)(2,9,7,13)(3,10,12,14)(5,8,16,11)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,36,38,45)(34,40,42,48)(35,44,43,37)(39,47,46,41);(1,5,6,16)(2,10,7,14)(3,9,12,13)(4,8,15,11)(17,22,23,26)(18,25,20,19)(21,29,30,32)(24,31,28,27)(33,37,38,44)(34,41,42,47)(35,45,43,36)(39,48,46,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=i,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8241735);