#8479566

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

4030 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8386762);