#8734566

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

4122 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8641762);