#8898580

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

4233 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)(3,5)(4,11)(6,14)(7,9)(8,15)(10,12)(13,16)(17,18,24,28)(19,29,32,25)(20,26,21,23)(22,27,31,30)(33,34)(35,44)(36,47)(37,48)(38,42)(39,40)(41,46)(43,45);(1,3,7,11)(2,5,9,4)(6,10,15,16)(8,13,14,12)(17,19)(18,25)(20,27)(21,30)(22,26)(23,31)(24,32)(28,29)(33,35,40,46)(34,41,39,44)(36,45,38,37)(42,43,47,48);(1,4,7,5)(2,3,9,11)(6,13,15,12)(8,16,14,10)(17,20,24,21)(18,26,28,23)(19,27,32,30)(22,29,31,25)(33,36,40,38)(34,42,39,47)(35,45,46,37)(41,43,44,48);(1,5,7,4)(2,3,9,11)(6,13,15,12)(8,10,14,16)(17,21,24,20)(18,26,28,23)(19,30,32,27)(22,29,31,25)(33,37)(34,43)(35,36)(38,46)(39,48)(40,45)(41,47)(42,44);(1,3)(2,5)(4,9)(6,10)(7,11)(8,13)(12,14)(15,16)(17,22,24,31)(18,27,28,30)(19,26,32,23)(20,29,21,25)(33,38,40,36)(34,42,39,47)(35,45,46,37)(41,48,44,43);(1,6)(2,8)(3,10)(4,12)(5,13)(7,15)(9,14)(11,16)(17,23,24,26)(18,20,28,21)(19,31,32,22)(25,27,29,30)(33,39,40,34)(35,44,46,41)(36,42,38,47)(37,48,45,43)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805776);