#8813971

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	#44005	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4015 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gi,d^2=h,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, 8721167);