#8343512

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

3908 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=ghi,d^2=g,e^2=gi,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, 8250708);