#10166802

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	#2675	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3974 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=ghi,e^2=i,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, 10073998);