#8662921

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

4174 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)(4,9)(5,6)(7,11)(8,15)(12,14)(13,16)(17,18)(19,25)(20,22)(21,24)(23,27)(26,32)(28,30)(29,31)(33,34,39,42)(35,38,40,36)(37,41,46,48)(43,47,45,44);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,16)(10,11)(18,20)(23,25)(24,28)(31,32)(33,35)(34,38)(36,42)(37,45)(39,40)(41,44)(43,46)(47,48);(1,4,7,5)(2,9,11,6)(3,12,15,13)(8,16,10,14)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,32)(29,30)(33,36,39,38)(34,40,42,35)(37,44,46,47)(41,45,48,43);(1,5,7,4)(2,9,11,6)(3,13,15,12)(8,16,10,14)(17,20,22,18)(19,25,27,23)(21,24,30,28)(26,31,29,32)(33,37)(34,41)(35,43)(36,44)(38,47)(39,46)(40,45)(42,48);(1,3)(2,10)(4,12)(5,13)(6,16)(7,15)(8,11)(9,14)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,30)(28,31)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,46)(47,48);(1,6,7,9)(2,5,11,4)(3,14,15,16)(8,13,10,12)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,38,39,36)(34,40,42,35)(37,44,46,47)(41,43,48,45)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8570117);