#8734596

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

4117 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)(3,12,6,8)(4,10,14,16)(5,13,15,11)(17,18)(19,23)(20,29)(21,25)(22,26)(24,28)(27,32)(30,31)(33,34)(35,41)(36,40)(37,42)(38,43)(39,46)(44,45)(47,48);(1,3,7,6)(2,8,9,12)(4,11,14,13)(5,10,15,16)(17,19,22,27)(18,23,26,32)(20,21,28,31)(24,30,29,25)(33,35)(34,39)(36,44)(37,45)(38,46)(40,47)(41,43)(42,48);(2,9)(4,14)(8,12)(11,13)(19,27)(20,28)(23,32)(24,29)(33,36,38,42)(34,40,43,37)(35,44,46,48)(39,47,41,45);(1,4,7,14)(2,10,9,16)(3,13,6,11)(5,12,15,8)(17,20)(18,24)(19,21)(22,28)(23,30)(25,32)(26,29)(27,31)(33,36)(34,40)(35,44)(37,43)(38,42)(39,47)(41,45)(46,48);(1,5)(2,11)(3,10)(4,12)(6,16)(7,15)(8,14)(9,13)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,32)(28,29)(33,35)(34,41)(36,44)(37,47)(38,46)(39,43)(40,45)(42,48);(1,6,7,3)(2,8,9,12)(4,13,14,11)(5,10,15,16)(17,21)(18,25)(19,28)(20,27)(22,31)(23,24)(26,30)(29,32)(33,37,38,40)(34,42,43,36)(35,45,46,47)(39,48,41,44)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,d^2=h,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, 8641792);