#8517584

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
prime radical	#52351	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3974 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)(3,9)(4,6)(5,8)(7,11)(10,16)(12,14)(13,15)(33,34,39,35)(36,43,46,40)(37,41,38,42)(44,48,45,47);(1,3,6,11)(2,7,4,9)(5,10,14,13)(8,15,12,16)(17,18,22,23)(19,24,29,32)(20,25,30,28)(21,26,31,27)(34,35)(40,43)(41,42)(47,48);(2,4)(7,9)(8,12)(15,16)(18,23)(20,30)(24,32)(26,27)(33,35,39,34)(36,43,46,40)(37,42,38,41)(44,48,45,47);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,14)(12,15)(17,19)(18,24)(20,27)(21,25)(22,29)(23,32)(26,30)(28,31)(33,36)(34,40)(35,43)(37,44)(38,45)(39,46)(41,47)(42,48);(1,4,6,2)(3,9,11,7)(5,12,14,8)(10,16,13,15)(17,20)(18,25)(19,27)(21,24)(22,30)(23,28)(26,29)(31,32)(33,37,39,38)(34,41,35,42)(36,44,46,45)(40,47,43,48);(1,5,6,14)(2,8,4,12)(3,10,11,13)(7,15,9,16)(17,21)(18,26)(19,28)(20,24)(22,31)(23,27)(25,29)(30,32)(33,38,39,37)(34,42,35,41)(36,45,46,44)(40,48,43,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,e^2=g,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, 8424780);