#8184789

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

4177 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)(3,8)(4,12)(5,10)(6,7)(9,14)(11,13)(15,16)(17,18)(19,20)(21,30)(22,25)(23,26)(24,27)(28,31)(29,32)(33,34,40,37)(35,41,38,47)(36,45,44,42)(39,43,46,48);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,14)(13,16)(17,19)(18,21)(20,27)(22,28)(23,29)(24,30)(25,26)(31,32)(33,35,40,38)(34,41,37,47)(36,39,44,46)(42,48,45,43);(2,7)(5,8)(9,12)(13,15)(17,20,24,21)(18,19,27,30)(22,31,29,26)(23,25,28,32)(33,36)(34,42)(35,39)(37,45)(38,46)(40,44)(41,48)(43,47);(1,3,6,10)(2,8,7,5)(4,11,14,16)(9,15,12,13)(17,21,24,20)(18,19,27,30)(22,26,29,31)(23,25,28,32)(33,37,40,34)(35,41,38,47)(36,45,44,42)(39,48,46,43);(1,4,6,14)(2,9,7,12)(3,11,10,16)(5,13,8,15)(17,22,24,29)(18,25,27,32)(19,28,30,23)(20,31,21,26)(33,38,40,35)(34,41,37,47)(36,46,44,39)(42,48,45,43);(1,5,6,8)(2,10,7,3)(4,13,14,15)(9,16,12,11)(17,23,24,28)(18,26,27,31)(19,29,30,22)(20,32,21,25)(33,39,40,46)(34,43,37,48)(35,44,38,36)(41,42,47,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=g,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8091985);