#10159202

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

4095 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)(4,7)(6,12)(8,10)(9,14)(11,15)(13,16)(17,18)(19,21)(20,24)(22,26)(23,30)(25,31)(27,32)(28,29)(33,34,40,44)(35,38,41,36)(37,47,46,42)(39,48,45,43);(1,3)(2,8)(4,5)(6,11)(7,10)(9,15)(12,13)(14,16)(17,19,22,25)(18,21,26,31)(20,28,30,27)(23,32,24,29)(33,35)(34,36)(37,45)(38,44)(39,46)(40,41)(42,43)(47,48);(2,4)(3,10)(9,12)(11,16)(17,20,22,30)(18,23,26,24)(19,27,25,28)(21,29,31,32)(33,36,40,38)(34,41,44,35)(37,48,46,43)(39,42,45,47);(1,4,7,2)(3,5,10,8)(6,12,14,9)(11,13,16,15)(17,20)(18,24)(19,28)(21,29)(22,30)(23,26)(25,27)(31,32)(33,37,40,46)(34,42,44,47)(35,45,41,39)(36,43,38,48);(1,5)(2,3)(4,10)(6,13)(7,8)(9,11)(12,16)(14,15)(17,19)(18,21)(20,27)(22,25)(23,29)(24,32)(26,31)(28,30)(33,38,40,36)(34,41,44,35)(37,43,46,48)(39,42,45,47);(1,6,7,14)(2,9,4,12)(3,11,10,16)(5,13,8,15)(17,21)(18,25)(19,26)(20,29)(22,31)(23,28)(24,27)(30,32)(33,39,40,45)(34,43,44,48)(35,46,41,37)(36,42,38,47)

Polycyclic presentation:

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

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, 10066398);