#8696802

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	#54780	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4022 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gi,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8603998);