#8399496

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	#59860	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4049 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)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,18,23,26)(19,22,25,32)(20,21,24,31)(27,28,29,30)(33,34)(35,42)(36,41)(37,46)(38,44)(39,40)(43,47)(45,48);(2,3)(4,11)(8,10)(12,16)(17,19)(18,22)(20,29)(21,30)(23,25)(24,27)(26,32)(28,31)(34,40)(37,47)(41,42)(44,45);(5,13)(8,10)(12,16)(14,15)(19,25)(22,32)(27,29)(28,30)(33,35,39,36)(34,41,40,42)(37,44,47,45)(38,46,48,43);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,20)(18,21)(19,27)(22,28)(23,24)(25,29)(26,31)(30,32)(33,36,39,35)(34,42,40,41)(37,44,47,45)(38,46,48,43);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,21)(18,24)(19,28)(20,26)(22,29)(23,31)(25,30)(27,32)(33,37,39,47)(34,43,40,46)(35,44,36,45)(38,42,48,41);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,22,23,32)(18,25,26,19)(20,28,24,30)(21,29,31,27)(33,38,39,48)(34,44,40,45)(35,43,36,46)(37,42,47,41)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8306692);