#8159974

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

4234 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=hi,e^2=h,f^2=g,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, 8067170);