#10142141

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
prime radical	#55560	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3905 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)(3,6)(4,9)(5,10)(7,8)(11,16)(12,14)(13,15)(17,18)(19,27)(20,30)(21,26)(22,24)(23,25)(28,32)(29,31)(33,34,40,38)(35,43,39,41)(36,42,37,46)(44,48,45,47);(1,3,7,11)(2,6,8,16)(4,12,13,5)(9,14,15,10)(17,19,23,22)(18,24,25,27)(20,28,26,31)(21,32,30,29)(33,35,40,39)(34,41,38,43)(36,44,37,45)(42,47,46,48);(2,8)(3,11)(4,13)(10,14)(18,25)(20,26)(24,27)(28,31)(33,36,40,37)(34,42,38,46)(35,44,39,45)(41,47,43,48);(1,4)(2,9)(3,12)(5,11)(6,14)(7,13)(8,15)(10,16)(17,20)(18,21)(19,28)(22,31)(23,26)(24,32)(25,30)(27,29)(33,37,40,36)(34,42,38,46)(35,45,39,44)(41,47,43,48);(1,5,7,12)(2,10,8,14)(3,13,11,4)(6,15,16,9)(17,21)(18,26)(19,29)(20,25)(22,32)(23,30)(24,28)(27,31)(33,38,40,34)(35,43,39,41)(36,46,37,42)(44,48,45,47);(1,6)(2,11)(3,8)(4,14)(5,9)(7,16)(10,13)(12,15)(17,22,23,19)(18,27,25,24)(20,31,26,28)(21,29,30,32)(33,39,40,35)(34,41,38,43)(36,45,37,44)(42,47,46,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=ghi,d^2=ghi,e^2=gh,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, 10049337);