#8399500

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

4146 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)(17,18,23,26)(19,22,25,32)(20,21,24,31)(27,28,29,30)(33,34,39,37)(35,40,36,42)(38,41,46,45)(43,47,44,48);(2,7)(4,10)(8,11)(12,15)(17,19)(18,22)(20,29)(21,30)(23,25)(24,27)(26,32)(28,31)(33,34)(35,40)(36,42)(37,39)(38,41)(43,47)(44,48)(45,46);(2,7)(5,9)(8,11)(13,16)(19,25)(22,32)(27,29)(28,30)(33,35,39,36)(34,40,37,42)(38,44,46,43)(41,48,45,47);(1,3,6,14)(2,8,7,11)(4,15,10,12)(5,13,9,16)(17,20)(18,21)(19,27)(22,28)(23,24)(25,29)(26,31)(30,32)(33,36,39,35)(34,40,37,42)(38,44,46,43)(41,47,45,48);(1,4,6,10)(2,9,7,5)(3,12,14,15)(8,13,11,16)(17,21)(18,24)(19,28)(20,26)(22,29)(23,31)(25,30)(27,32)(33,37,39,34)(35,42,36,40)(38,45,46,41)(43,48,44,47);(1,5,6,9)(2,10,7,4)(3,13,14,16)(8,12,11,15)(17,22,23,32)(18,25,26,19)(20,28,24,30)(21,29,31,27)(33,38,39,46)(34,41,37,45)(35,43,36,44)(40,47,42,48)

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,d^2=g,e^2=g,f^2=gi,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, 8306696);