#8533592

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

4026 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=hi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8440788);