#9230869

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	#44445	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4090 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=h,e^2=ghi,f^2=h,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, 9138065);