#10246842

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

4152 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10154038);