#8687329

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

3956 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=gh,d^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,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, 8594525);