#8159026

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

4049 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)(5,7)(6,13)(8,10)(9,14)(11,16)(12,15)(17,18,23,24)(19,20,21,25)(22,30,31,26)(27,32,29,28)(33,34,39,42)(35,43,46,40)(36,47,45,37)(38,48,44,41);(1,3,7,8)(2,4,5,10)(6,11,14,15)(9,12,13,16)(18,24)(20,25)(26,30)(28,32)(33,35)(34,40)(36,44)(37,41)(38,45)(39,46)(42,43)(47,48);(1,4,7,10)(2,8,5,3)(6,12,14,16)(9,15,13,11)(17,19,23,21)(18,25,24,20)(22,27,31,29)(26,32,30,28)(33,34,39,42)(35,40,46,43)(36,47,45,37)(38,41,44,48);(1,4)(2,3)(5,8)(6,12)(7,10)(9,11)(13,15)(14,16)(17,20,23,25)(18,19,24,21)(22,28,31,32)(26,27,30,29)(33,36,39,45)(34,37,42,47)(35,44,46,38)(40,41,43,48);(1,5,7,2)(3,10,8,4)(6,13,14,9)(11,16,15,12)(17,21,23,19)(18,20,24,25)(22,29,31,27)(26,28,30,32)(33,37)(34,36)(35,41)(38,43)(39,47)(40,44)(42,45)(46,48);(1,6,7,14)(2,9,5,13)(3,11,8,15)(4,12,10,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,30)(25,32)(33,38,39,44)(34,41,42,48)(35,45,46,36)(37,43,47,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,d^2=g,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8066222);