#8127224

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

4152 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)(3,7)(4,12)(5,13)(6,8)(9,14)(10,16)(11,15)(17,18)(19,28)(20,25)(21,26)(22,32)(23,27)(24,31)(29,30)(37,45)(39,46)(42,48)(44,47);(1,3,6,9)(2,7,8,14)(4,10,13,15)(5,11,12,16)(17,19,23,31)(18,24,27,28)(20,29,32,21)(22,30,25,26)(33,34,38,40)(35,41,43,36)(37,39,45,46)(42,47,48,44);(2,8)(3,9)(4,13)(11,16)(17,20,23,32)(18,25,27,22)(19,29,31,21)(24,26,28,30)(33,35,38,43)(34,36,40,41)(37,42,45,48)(39,44,46,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,20)(18,22)(19,29)(21,31)(23,32)(24,30)(25,27)(26,28)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,47)(46,48);(1,4)(2,5)(3,10)(6,13)(7,11)(8,12)(9,15)(14,16)(17,21,23,29)(18,26,27,30)(19,20,31,32)(22,28,25,24)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,47)(46,48);(1,5)(2,4)(3,11)(6,12)(7,10)(8,13)(9,16)(14,15)(17,22)(18,20)(19,30)(21,28)(23,25)(24,29)(26,31)(27,32)(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,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, 8034420);