#8664376

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

4126 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)(3,11,13,8)(4,14,12,9)(5,10,15,16)(17,18,24,23)(19,25,22,29)(20,30,21,26)(27,32,28,31)(33,34,38,39)(35,41,36,40)(37,42,46,45)(43,48,44,47);(1,3,7,13)(2,8,6,11)(4,5,12,15)(9,10,14,16)(17,19,24,22)(18,25,23,29)(20,28,21,27)(26,32,30,31)(34,39)(40,41)(42,45)(47,48);(17,20,24,21)(18,26,23,30)(19,27,22,28)(25,31,29,32)(33,35,38,36)(34,40,39,41)(37,43,46,44)(42,47,45,48);(1,4,7,12)(2,9,6,14)(3,5,13,15)(8,10,11,16)(17,21,24,20)(18,26,23,30)(19,28,22,27)(25,31,29,32)(33,34)(35,40)(36,41)(37,45)(38,39)(42,46)(43,48)(44,47);(1,5)(2,10)(3,12)(4,13)(6,16)(7,15)(8,14)(9,11)(17,22,24,19)(18,25,23,29)(20,28,21,27)(26,31,30,32)(33,36,38,35)(34,41,39,40)(37,43,46,44)(42,47,45,48);(1,6,7,2)(3,11,13,8)(4,9,12,14)(5,10,15,16)(17,23,24,18)(19,29,22,25)(20,30,21,26)(27,32,28,31)(33,37)(34,42)(35,43)(36,44)(38,46)(39,45)(40,47)(41,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8571572);