#8534803

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

4089 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=g,e^2=ghi,f^2=g,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, 8441999);