#8393356

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8300552);