#10504574

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

4163 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=g,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10411770);