#8630825

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

3979 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)(3,10)(4,12)(5,6)(7,8)(9,15)(11,14)(13,16)(17,18,23,25)(19,24,29,31)(20,26,27,21)(22,28,32,30)(33,34,39,40)(35,41,36,38)(37,45,47,42)(43,46,44,48);(1,3,7,6)(2,5,8,10)(4,13,15,11)(9,16,12,14)(17,19)(18,24)(20,30)(21,32)(22,26)(23,29)(25,31)(27,28)(34,40)(35,36)(42,45)(43,44);(2,8)(5,10)(9,12)(14,16)(18,25)(19,29)(21,26)(28,30)(33,35,39,36)(34,38,40,41)(37,43,47,44)(42,46,45,48);(1,4,7,15)(2,9,8,12)(3,11,6,13)(5,14,10,16)(17,20,23,27)(18,26,25,21)(19,28,29,30)(22,24,32,31)(33,36,39,35)(34,41,40,38)(37,44,47,43)(42,48,45,46);(1,5)(2,3)(4,14)(6,8)(7,10)(9,11)(12,13)(15,16)(17,21,23,26)(18,27,25,20)(19,22,29,32)(24,30,31,28)(33,37,39,47)(34,42,40,45)(35,43,36,44)(38,46,41,48);(1,6,7,3)(2,5,8,10)(4,11,15,13)(9,16,12,14)(17,22)(18,28)(19,26)(20,31)(21,29)(23,32)(24,27)(25,30)(33,38,39,41)(34,35,40,36)(37,46,47,48)(42,43,45,44)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=ghi,d^2=h,e^2=i,f^2=g,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, 8538021);