#8348914

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

4244 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)(3,10)(4,12)(5,9)(6,7)(8,13)(11,15)(14,16)(17,18,23,27)(19,24,28,32)(20,21,25,30)(22,26,31,29)(33,34,39,37)(35,41,36,40)(38,42,46,45)(43,48,44,47);(2,7)(3,9)(4,13)(15,16)(17,19)(18,24)(20,22)(21,26)(23,28)(25,31)(27,32)(29,30)(33,35,39,36)(34,40,37,41)(38,43,46,44)(42,47,45,48);(2,7)(5,10)(8,12)(15,16)(19,28)(22,31)(24,32)(26,29)(33,35,39,36)(34,41,37,40)(38,44,46,43)(42,47,45,48);(1,3,6,9)(2,5,7,10)(4,14,13,11)(8,16,12,15)(17,20)(18,21)(19,22)(23,25)(24,26)(27,30)(28,31)(29,32)(33,36,39,35)(34,41,37,40)(38,44,46,43)(42,48,45,47);(1,4,6,13)(2,8,7,12)(3,11,9,14)(5,15,10,16)(17,21)(18,25)(19,29)(20,27)(22,24)(23,30)(26,28)(31,32)(33,37,39,34)(35,40,36,41)(38,45,46,42)(43,47,44,48);(1,5)(2,9)(3,7)(4,15)(6,10)(8,14)(11,12)(13,16)(17,22)(18,26)(19,20)(21,24)(23,31)(25,28)(27,29)(30,32)(33,38)(34,42)(35,43)(36,44)(37,45)(39,46)(40,47)(41,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=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, 8256110);