#9668307

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	#52383	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3879 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)(3,9)(4,8)(5,10)(6,11)(7,12)(13,14)(15,16)(17,18)(19,25)(20,28)(21,24)(22,27)(23,30)(26,31)(29,32)(33,34,39,36)(35,40,45,43)(37,41,44,46)(38,42,48,47);(1,3,7,4)(2,8,12,9)(5,13,11,15)(6,14,10,16)(17,18,21,24)(19,22,27,25)(20,23,30,28)(26,31,29,32)(33,35,39,45)(34,40,36,43)(37,38,44,48)(41,42,46,47);(1,3,7,4)(2,9,12,8)(5,15,11,13)(6,14,10,16)(17,19,21,27)(18,22,24,25)(20,26,30,29)(23,31,28,32)(33,34,39,36)(35,43,45,40)(37,46,44,41)(38,42,48,47);(1,4,7,3)(2,9,12,8)(5,13,11,15)(6,14,10,16)(17,19)(18,22)(20,29)(21,27)(23,32)(24,25)(26,30)(28,31)(33,36,39,34)(35,43,45,40)(37,41,44,46)(38,42,48,47);(1,5,7,11)(2,10,12,6)(3,13,4,15)(8,16,9,14)(17,18)(19,25)(20,23)(21,24)(22,27)(26,32)(28,30)(29,31)(33,37,39,44)(34,41,36,46)(35,38,45,48)(40,42,43,47);(1,6,7,10)(2,11,12,5)(3,14,4,16)(8,15,9,13)(17,20)(18,23)(19,26)(21,30)(22,31)(24,28)(25,32)(27,29)(33,38)(34,42)(35,44)(36,47)(37,45)(39,48)(40,46)(41,43)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9575503);