#9206544

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	#53735	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3964 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,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, 9113740);