#9508911

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3851 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9416107);