#8323302

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3920 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=g,f^2=gi,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, 8230498);