#8696729

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

4128 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8603925);