#9203857

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3903 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=g,e^2=gi,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, 9111053);