#9009234

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3928 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8916430);