#8339983

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

4051 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)(3,5,10,8)(4,12,15,9)(11,16,13,14)(17,18,21,24)(19,22,27,25)(20,23,28,30)(26,29,32,31)(33,34,39,36)(35,40,38,42)(37,41,46,44)(43,47,45,48);(2,7)(3,10)(9,12)(11,13)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(2,7)(5,8)(9,12)(14,16)(20,28)(23,30)(26,32)(29,31)(33,34,39,36)(35,42,38,40)(37,41,46,44)(43,48,45,47);(1,3,6,10)(2,8,7,5)(4,13,15,11)(9,14,12,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,36,39,34)(35,42,38,40)(37,41,46,44)(43,47,45,48);(1,4,6,15)(2,9,7,12)(3,11,10,13)(5,14,8,16)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,5,6,8)(2,10,7,3)(4,14,15,16)(9,13,12,11)(17,20)(18,23)(19,26)(21,28)(22,29)(24,30)(25,31)(27,32)(33,38,39,35)(34,42,36,40)(37,45,46,43)(41,48,44,47)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8247179);