#8602482

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	#55253	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4047 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8509678);