#8944465

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

4166 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=hi,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=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, 8851661);