#9175012

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	#50480	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4031 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)(3,7)(4,10)(5,8)(6,9)(11,15)(12,16)(13,14)(18,23)(19,25)(21,29)(28,30)(33,34,39,44)(35,45,47,40)(36,41,43,38)(37,42,48,46);(1,3,6,10)(2,7,9,4)(5,11,14,16)(8,15,13,12)(18,23)(20,26)(24,31)(28,30)(33,34)(35,40)(36,41)(37,46)(38,43)(39,44)(42,48)(45,47);(3,10)(4,7)(11,16)(12,15)(17,18,22,23)(19,26,25,20)(21,24,29,31)(27,30,32,28)(33,35,39,47)(34,40,44,45)(36,48,43,37)(38,46,41,42);(1,4)(2,3)(5,12)(6,7)(8,11)(9,10)(13,16)(14,15)(17,19,22,25)(18,20,23,26)(21,27,29,32)(24,28,31,30)(33,36,39,43)(34,41,44,38)(35,37,47,48)(40,46,45,42);(1,2)(3,7)(4,10)(5,13)(6,9)(8,14)(11,12)(15,16)(17,20)(18,19)(21,30)(22,26)(23,25)(24,32)(27,31)(28,29)(33,37)(34,42)(35,36)(38,40)(39,48)(41,45)(43,47)(44,46);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,21)(18,24)(19,27)(20,28)(22,29)(23,31)(25,32)(26,30)(33,38,39,41)(34,43,44,36)(35,46,47,42)(37,45,48,40)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9082208);