#8810081

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	#51126	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4069 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=g,e^2=h,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, 8717277);