#8843592

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

4118 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:

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

Polycyclic presentation:

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

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, 8750788);