#9214734

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	#2541	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4080 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=g,e^2=g,f^2=g,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, 9121930);