#8110439

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

4138 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:

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

Polycyclic presentation:

2^9|c^2=g,d^2=gi,e^2=gi,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, 8017635);