#10106565

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

4249 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)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18,23,25)(19,22,20,24)(21,26,30,32)(27,29,28,31)(33,34)(35,36)(37,46)(38,41)(39,47)(40,44)(42,45)(43,48);(4,11)(8,14)(10,15)(12,16)(17,19,23,20)(18,24,25,22)(21,28,30,27)(26,29,32,31)(33,35,40,41)(34,38,44,36)(37,39,45,48)(42,47,46,43);(2,7)(3,9)(8,14)(10,15)(18,25)(19,20)(26,32)(27,28)(33,36,40,38)(34,41,44,35)(37,47,45,43)(39,46,48,42);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,20,23,19)(18,22,25,24)(21,28,30,27)(26,29,32,31)(33,37,40,45)(34,42,44,46)(35,39,41,48)(36,43,38,47);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,23,30)(18,26,25,32)(19,27,20,28)(22,29,24,31)(33,38,40,36)(34,41,44,35)(37,43,45,47)(39,46,48,42);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,22)(18,19)(20,25)(21,29)(23,24)(26,27)(28,32)(30,31)(33,39)(34,43)(35,45)(36,46)(37,41)(38,42)(40,48)(44,47)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=h,e^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, 10013761);