#8976393

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	#40941	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4118 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8883589);