#9668311

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	#52967	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4047 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)(3,8)(4,10)(5,11)(6,16)(7,9)(12,13)(14,15)(18,23)(19,25)(24,31)(26,28)(33,34,38,42)(35,36,40,39)(37,45,47,41)(43,48,46,44);(1,3,7,6)(2,8,9,16)(4,12,11,14)(5,15,10,13)(17,18,22,23)(19,21,25,30)(20,24,27,31)(26,29,28,32)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,48)(46,47);(2,9)(4,11)(8,16)(12,14)(20,27)(24,31)(26,28)(29,32)(33,35,38,40)(34,39,42,36)(37,43,47,46)(41,48,45,44);(1,4,7,11)(2,10,9,5)(3,12,6,14)(8,13,16,15)(17,19)(18,21)(20,28)(22,25)(23,30)(24,32)(26,27)(29,31)(33,35)(34,36)(37,46)(38,40)(39,42)(41,48)(43,47)(44,45);(1,5,7,10)(2,11,9,4)(3,13,6,15)(8,12,16,14)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,36)(34,40)(35,42)(37,44)(38,39)(41,46)(43,45)(47,48);(1,6,7,3)(2,8,9,16)(4,14,11,12)(5,15,10,13)(17,21)(18,25)(19,23)(20,29)(22,30)(24,28)(26,31)(27,32)(33,37,38,47)(34,41,42,45)(35,43,40,46)(36,44,39,48)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=i,d^2=hi,e^2=hi,f^2=h,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, 9575507);