#8993348

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	#50053	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3991 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)(3,8)(4,9)(5,14)(6,7)(10,16)(11,15)(12,13)(17,18,22,23)(19,27,30,24)(20,25,31,29)(21,28,32,26)(33,34,38,37)(35,39,44,42)(36,40,45,46)(41,47,43,48);(1,2)(3,8)(4,12)(5,10)(6,7)(9,13)(11,15)(14,16)(18,23)(20,31)(24,27)(26,28)(33,35)(34,39)(36,43)(37,42)(38,44)(40,48)(41,45)(46,47);(2,7)(3,11)(4,13)(5,16)(17,19,22,30)(18,24,23,27)(20,26,31,28)(21,25,32,29)(33,36,38,45)(34,40,37,46)(35,41,44,43)(39,47,42,48);(1,3,6,11)(2,8,7,15)(4,14,13,10)(5,12,16,9)(17,19)(18,24)(20,26)(21,29)(22,30)(23,27)(25,32)(28,31)(33,37,38,34)(35,42,44,39)(36,40,45,46)(41,47,43,48);(1,4)(2,9)(3,10)(5,15)(6,13)(7,12)(8,16)(11,14)(17,20)(18,25)(19,28)(21,24)(22,31)(23,29)(26,30)(27,32)(33,36)(34,40)(35,43)(37,46)(38,45)(39,48)(41,44)(42,47);(1,5,6,16)(2,10,7,14)(3,12,11,9)(4,15,13,8)(17,21)(18,26)(19,29)(20,24)(22,32)(23,28)(25,30)(27,31)(33,35)(34,39)(36,43)(37,42)(38,44)(40,48)(41,45)(46,47)

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^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, 8900544);