#9230885

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	#48403	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4018 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=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, 9138081);