#8551187

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

4265 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)(17,18,22,23)(19,20,27,25)(21,24,30,31)(26,28,29,32)(33,34)(35,43)(36,45)(37,39)(38,42)(40,44)(41,47)(46,48);(1,2,6,3)(4,7,11,9)(5,8,13,10)(12,14,16,15)(17,18,22,23)(19,25,27,20)(21,24,30,31)(26,32,29,28)(33,35,39,44)(34,40,37,43)(36,46,47,38)(41,48,45,42);(4,11)(7,9)(12,16)(14,15)(18,23)(20,25)(24,31)(28,32)(34,37)(40,43)(41,45)(42,48);(1,3,6,2)(4,9,11,7)(5,8,13,10)(12,14,16,15)(17,19)(18,20)(21,29)(22,27)(23,25)(24,32)(26,30)(28,31)(33,36,39,47)(34,41,37,45)(35,38,44,46)(40,42,43,48);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,31)(30,32)(33,37,39,34)(35,43,44,40)(36,45,47,41)(38,48,46,42);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,38)(34,42)(35,36)(37,48)(39,46)(40,41)(43,45)(44,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,d^2=g,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8458383);