#8449297

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

4106 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)(3,12)(4,6)(5,15)(7,11)(8,16)(9,14)(10,13)(18,22)(19,25)(24,29)(26,31)(33,34)(35,39)(36,40)(37,48)(38,42)(41,44)(43,45)(46,47);(1,3)(2,8)(4,13)(5,14)(6,15)(7,16)(9,10)(11,12)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,34)(35,43)(36,40)(37,41)(38,42)(39,45)(44,48)(46,47);(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,35,38,45)(34,39,42,43)(36,44,47,37)(40,41,46,48);(1,4,7,14)(2,9,11,6)(3,13,16,5)(8,10,12,15)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,36)(34,40)(35,44)(37,45)(38,47)(39,41)(42,46)(43,48);(1,5,7,13)(2,10,11,15)(3,14,16,4)(6,8,9,12)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,37,38,44)(34,41,42,48)(35,36,45,47)(39,46,43,40);(1,6,7,9)(2,4,11,14)(3,15,16,10)(5,8,13,12)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,34)(35,39)(36,46)(37,41)(38,42)(40,47)(43,45)(44,48)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8356493);