#8127591

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	#39580	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3995 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=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, 8034787);