#8549523

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

4286 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)(34,39)(35,36)(41,46)(43,44);(1,2,6,3)(4,7,11,9)(5,8,13,10)(12,14,16,15)(17,18,22,23)(19,24,28,31)(20,25,29,21)(26,27,32,30)(33,34)(35,40)(36,42)(37,41)(38,39)(43,47)(44,48)(45,46);(4,11)(7,9)(12,16)(14,15)(18,23)(20,29)(24,31)(26,32)(33,35,38,36)(34,40,39,42)(37,43,45,44)(41,47,46,48);(1,3,6,2)(4,9,11,7)(5,8,13,10)(12,14,16,15)(17,19)(18,24)(20,26)(21,30)(22,28)(23,31)(25,27)(29,32)(33,36,38,35)(34,40,39,42)(37,43,45,44)(41,48,46,47);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,20)(18,25)(19,26)(21,23)(22,29)(24,27)(28,32)(30,31)(33,34)(35,42)(36,40)(37,41)(38,39)(43,48)(44,47)(45,46);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,21,22,25)(18,20,23,29)(19,27,28,30)(24,32,31,26)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|b^2=ghi,c^2=h,d^2=g,e^2=gh,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, 8456719);