#8824126

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	#57488	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3984 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)(3,6)(4,12)(5,8)(7,14)(9,11)(10,16)(13,15)(17,18,23,20)(19,26,22,24)(21,25,29,28)(27,32,30,31)(33,34)(35,43)(36,37)(38,39)(40,46)(41,42)(44,45)(47,48);(1,3,5,11)(2,6,8,9)(4,13,14,10)(7,16,12,15)(21,29)(25,28)(27,30)(31,32)(34,39)(37,41)(40,43)(45,48);(1,2,5,8)(3,9,11,6)(4,7,14,12)(10,16,13,15)(17,19,23,22)(18,24,20,26)(21,27,29,30)(25,31,28,32)(33,35,38,46)(34,40,39,43)(36,47,42,44)(37,45,41,48);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,14)(13,16)(17,20,23,18)(19,26,22,24)(21,28,29,25)(27,32,30,31)(33,36,38,42)(34,41,39,37)(35,44,46,47)(40,45,43,48);(1,3)(2,6)(4,10)(5,11)(7,15)(8,9)(12,16)(13,14)(17,21,23,29)(18,25,20,28)(19,27,22,30)(24,31,26,32)(33,34)(35,43)(36,41)(37,42)(38,39)(40,46)(44,48)(45,47);(1,4)(2,7)(3,10)(5,14)(6,15)(8,12)(9,16)(11,13)(17,22,23,19)(18,26,20,24)(21,30,29,27)(25,32,28,31)(33,37,38,41)(34,42,39,36)(35,45,46,48)(40,44,43,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=h,d^2=hi,e^2=ghi,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, 8731322);