#8127222

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4264 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,e^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, 8034418);