#8926420

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

3983 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833616);