#8258633

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

4067 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:

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

Polycyclic presentation:

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

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, 8165829);