#8127355

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

4268 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)(19,24)(20,26)(21,31)(22,32)(23,25)(27,30)(28,29)(37,45)(39,46)(42,48)(44,47);(2,7)(3,10)(8,12)(11,16)(17,19)(18,24)(20,23)(21,27)(22,28)(25,26)(29,32)(30,31)(33,34,38,40)(35,41,43,36)(37,39,45,46)(42,47,48,44);(2,7)(3,10)(4,13)(14,15)(18,25)(19,20)(21,32)(28,30)(33,35,38,43)(34,36,40,41)(37,42,45,48)(39,44,46,47);(1,3)(2,5)(4,11)(6,10)(7,9)(8,14)(12,15)(13,16)(17,20,23,19)(18,26,25,24)(21,29,32,27)(22,30,31,28)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,47)(46,48);(1,4,6,13)(2,8,7,12)(3,11,10,16)(5,14,9,15)(17,21)(18,22)(19,27)(20,29)(23,32)(24,28)(25,31)(26,30)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,47)(46,48);(1,5)(2,3)(4,14)(6,9)(7,10)(8,11)(12,16)(13,15)(17,22)(18,21)(19,28)(20,30)(23,31)(24,27)(25,32)(26,29)(33,37)(34,39)(35,42)(36,44)(38,45)(40,46)(41,47)(43,48)

Polycyclic presentation:

2^9|b^2=ghi,c^2=ghi,d^2=gh,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, 8034551);