#8533616

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

3955 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)(3,8)(4,11)(5,9)(6,7)(10,16)(12,13)(14,15)(17,18,23,22)(19,24,30,29)(20,26,21,25)(27,32,28,31)(33,34)(35,40)(36,44)(37,42)(38,39)(41,46)(43,45)(47,48);(1,2)(3,4)(5,9)(6,7)(8,11)(10,12)(13,16)(14,15)(17,19,23,30)(18,24,22,29)(20,27,21,28)(25,31,26,32)(34,39)(35,42)(41,44)(43,48);(2,7)(4,8)(9,15)(12,16)(17,20,23,21)(18,25,22,26)(19,27,30,28)(24,31,29,32)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,3)(2,8)(4,7)(5,13)(6,11)(9,12)(10,14)(15,16)(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,4,6,8)(2,3,7,11)(5,12,14,16)(9,10,15,13)(17,21,23,20)(18,26,22,25)(19,28,30,27)(24,32,29,31);(1,5,6,14)(2,9,7,15)(3,10,11,13)(4,12,8,16)(17,22,23,18)(19,29,30,24)(20,26,21,25)(27,32,28,31)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=hi,d^2=g,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, 8440812);