#8471143

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	#44086	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4101 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=ghi,d^2=i,e^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8378339);