#8449298

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

4114 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)(3,11,14,8)(4,15,13,9)(5,16,12,10)(17,18,22,21)(19,25,27,23)(20,28,29,24)(26,30,32,31)(33,34)(35,39)(36,40)(37,48)(38,42)(41,44)(43,45)(46,47);(1,3)(2,8)(4,12)(5,13)(6,11)(7,14)(9,10)(15,16)(17,19,22,27)(18,23,21,25)(20,26,29,32)(24,30,28,31)(33,34)(35,43)(36,40)(37,41)(38,42)(39,45)(44,48)(46,47);(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,35,38,45)(34,39,42,43)(36,44,47,37)(40,41,46,48);(1,4,7,13)(2,9,6,15)(3,12,14,5)(8,10,11,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,36)(34,40)(35,44)(37,45)(38,47)(39,41)(42,46)(43,48);(1,5,7,12)(2,10,6,16)(3,13,14,4)(8,9,11,15)(17,20,22,29)(18,24,21,28)(19,26,27,32)(23,30,25,31)(33,37,38,44)(34,41,42,48)(35,36,45,47)(39,46,43,40);(1,6,7,2)(3,11,14,8)(4,9,13,15)(5,16,12,10)(17,21,22,18)(19,25,27,23)(20,28,29,24)(26,31,32,30)(33,34)(35,39)(36,46)(37,41)(38,42)(40,47)(43,45)(44,48)

Polycyclic presentation:

2^9|a^2=g,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, 8356494);