#8695868

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

4177 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=ghi,d^2=gh,e^2=g,f^2=g,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, 8603064);