#8343518

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	#42685	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4004 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=hi,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, 8250714);