#8633675

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3948 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)(3,5,6,8)(4,12,15,10)(11,16,13,14)(33,34,38,41)(35,39,43,37)(36,44,47,40)(42,46,45,48);(1,3,7,6)(2,8,9,5)(4,13,15,11)(10,14,12,16)(17,18)(19,24)(20,21)(22,23)(25,29)(26,30)(27,32)(28,31)(33,35)(34,39)(36,45)(37,41)(38,43)(40,46)(42,47)(44,48);(2,9)(5,8)(10,12)(14,16)(18,23)(20,29)(24,31)(26,32)(33,36,38,47)(34,40,41,44)(35,42,43,45)(37,46,39,48);(1,4,7,15)(2,10,9,12)(3,11,6,13)(5,14,8,16)(17,19)(18,24)(20,26)(21,30)(22,28)(23,31)(25,27)(29,32)(33,37)(34,35)(36,46)(38,39)(40,42)(41,43)(44,45)(47,48);(1,5,7,8)(2,3,9,6)(4,14,15,16)(10,11,12,13)(17,20)(18,25)(19,26)(21,23)(22,29)(24,27)(28,32)(30,31)(33,36)(34,40)(35,42)(37,46)(38,47)(39,48)(41,44)(43,45);(1,6,7,3)(2,8,9,5)(4,11,15,13)(10,14,12,16)(17,21,22,25)(18,20,23,29)(19,27,28,30)(24,32,31,26)(33,35)(34,39)(36,42)(37,41)(38,43)(40,48)(44,46)(45,47)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=g,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8540871);