#8920756

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

4183 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,d^2=h,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8827952);