#8831179

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	#57180	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4274 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=i,d^2=g,e^2=g,f^2=gh,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, 8738375);