#8127336

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	#38521	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3931 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)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(17,18)(19,26)(20,24)(21,23)(22,27)(25,30)(28,32)(29,31)(37,45)(39,46)(42,48)(44,47);(2,3)(7,9)(8,10)(14,15)(17,19,22,21)(18,23,27,26)(20,28,30,29)(24,31,25,32)(33,34,38,40)(35,41,43,36)(37,39,45,46)(42,47,48,44);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,20,22,30)(18,24,27,25)(19,28,21,29)(23,31,26,32)(33,35,38,43)(34,36,40,41)(37,42,45,48)(39,44,46,47);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20)(18,25)(19,28)(21,29)(22,30)(23,32)(24,27)(26,31)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,47)(46,48);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,18)(19,23)(20,25)(21,26)(22,27)(24,30)(28,32)(29,31)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,47)(46,48);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21,22,19)(18,26,27,23)(20,29,30,28)(24,32,25,31)(33,37)(34,39)(35,42)(36,44)(38,45)(40,46)(41,47)(43,48)

Polycyclic presentation:

2^9|b^2=gi,c^2=ghi,d^2=h,f^2=g,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, 8034532);