#9029865

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	#42260	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3895 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8937061);