#8746089

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

3939 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)(7,9)(8,13)(10,12)(14,16)(18,22)(19,25)(24,27)(26,28)(33,34,40,37)(35,41,38,47)(36,45,44,42)(39,43,46,48);(1,3,7,13)(2,8,9,4)(5,6,12,16)(10,11,15,14)(18,22)(23,30)(24,27)(31,32)(33,35,40,38)(34,41,37,47)(36,39,44,46)(42,48,45,43);(2,9)(4,8)(10,15)(11,14)(17,18,21,22)(19,23,25,30)(20,24,29,27)(26,31,28,32)(33,36)(34,42)(35,39)(37,45)(38,46)(40,44)(41,48)(43,47);(1,4,7,8)(2,3,9,13)(5,14,12,11)(6,15,16,10)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(33,37,40,34)(35,41,38,47)(36,45,44,42)(39,48,46,43);(1,5)(2,10)(3,6)(4,14)(7,12)(8,11)(9,15)(13,16)(17,19,21,25)(18,23,22,30)(20,28,29,26)(24,32,27,31)(33,38,40,35)(34,41,37,47)(36,46,44,39)(42,48,45,43);(1,6)(2,11)(3,12)(4,10)(5,13)(7,16)(8,15)(9,14)(17,20,21,29)(18,24,22,27)(19,26,25,28)(23,31,30,32)(33,39,40,46)(34,43,37,48)(35,44,38,36)(41,42,47,45)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=i,d^2=hi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8653285);