#8831270

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	#48405	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4191 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8738466);