#9369126

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	#2825	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3750 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dghi,e^d=egi

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, 9276322);