#8806868

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	#40173	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4095 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=hi,d^2=hi,e^2=hi,f^2=h,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, 8714064);