#8994424

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	#43089	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4000 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=gi,e^2=h,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, 8901620);