#10407760

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

4122 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:

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

Polycyclic presentation:

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

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, 10314956);