#8110454

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017650);