#8464171

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4100 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)(3,7)(4,14)(5,9)(6,10)(8,11)(12,16)(13,15)(17,18,24,22)(19,29,30,25)(20,27,21,26)(23,28,31,32)(33,34,38,42)(35,39,40,36)(37,41,46,45)(43,47,48,44);(1,3,8,6)(2,7,11,10)(4,13,5,12)(9,16,14,15)(17,19,24,30)(18,25,22,29)(20,23,21,31)(26,28,27,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,48)(45,46);(1,4,8,5)(2,9,11,14)(3,12,6,13)(7,15,10,16)(17,20,24,21)(18,26,22,27)(19,23,30,31)(25,28,29,32)(33,35,38,40)(34,39,42,36)(37,43,46,48)(41,47,45,44);(1,5,8,4)(2,9,11,14)(3,13,6,12)(7,15,10,16)(17,21,24,20)(18,27,22,26)(19,23,30,31)(25,28,29,32)(33,34)(35,39)(36,40)(37,45)(38,42)(41,46)(43,44)(47,48);(1,6,8,3)(2,10,11,7)(4,13,5,12)(9,16,14,15)(17,22,24,18)(19,29,30,25)(20,27,21,26)(23,32,31,28)(33,36,38,39)(34,40,42,35)(37,44,46,47)(41,48,45,43);(1,7,8,10)(2,3,11,6)(4,15,5,16)(9,12,14,13)(17,23)(18,28)(19,21)(20,30)(22,32)(24,31)(25,27)(26,29)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,48)(42,45)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=h,e^2=gi,f^2=i,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, 8371367);