#8824118

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	#50043	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4131 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)(3,6)(4,11)(5,7)(8,13)(9,15)(10,16)(12,14)(17,18,22,23)(19,24,27,21)(20,25,30,31)(26,32,28,29)(33,34,39,36)(35,42,41,38)(37,40,45,44)(43,48,47,46);(1,3,5,9)(2,6,7,15)(4,12,13,10)(8,16,11,14)(18,23)(21,24)(25,31)(29,32)(37,45)(40,44)(43,47)(46,48);(2,7)(3,9)(8,11)(10,12)(17,19,22,27)(18,24,23,21)(20,28,30,26)(25,29,31,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,48)(45,47);(1,2)(3,6)(4,8)(5,7)(9,15)(10,14)(11,13)(12,16)(17,19)(18,21)(20,26)(22,27)(23,24)(25,29)(28,30)(31,32)(33,36,39,34)(35,42,41,38)(37,44,45,40)(43,48,47,46);(1,3)(2,6)(4,10)(5,9)(7,15)(8,14)(11,16)(12,13)(17,20,22,30)(18,25,23,31)(19,26,27,28)(21,29,24,32)(33,37,39,45)(34,40,36,44)(35,43,41,47)(38,46,42,48);(1,4)(2,8)(3,10)(5,13)(6,14)(7,11)(9,12)(15,16)(17,21)(18,19)(20,29)(22,24)(23,27)(25,26)(28,31)(30,32)(33,38,39,42)(34,41,36,35)(37,46,45,48)(40,47,44,43)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=g,d^2=hi,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8731314);