#9206574

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	#52407	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3983 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,d^2=gh,e^2=ghi,f^2=gh,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, 9113770);