#8688292

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

4237 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=i,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8595488);