#10040802

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	#2388	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3881 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)(3,10)(4,8)(5,13)(6,7)(9,15)(11,12)(14,16)(17,18)(19,22)(20,27)(21,24)(23,28)(25,29)(26,32)(30,31)(33,34,40,38)(35,41,37,46)(36,43,39,42)(44,48,45,47);(1,3,6,8)(2,4,7,10)(5,11,15,16)(9,12,13,14)(19,25)(22,29)(26,31)(30,32)(33,35)(34,41)(36,45)(37,40)(38,46)(39,44)(42,48)(43,47);(2,7)(4,10)(9,13)(12,14)(17,18,21,24)(19,22,25,29)(20,23,28,27)(26,30,31,32)(33,36,40,39)(34,42,38,43)(35,44,37,45)(41,47,46,48);(1,2)(3,10)(4,8)(5,9)(6,7)(11,14)(12,16)(13,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,37,40,35)(34,41,38,46)(36,44,39,45)(42,48,43,47);(1,4,6,10)(2,8,7,3)(5,14,15,12)(9,11,13,16)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32)(33,38,40,34)(35,41,37,46)(36,42,39,43)(44,48,45,47);(1,5,6,15)(2,9,7,13)(3,11,8,16)(4,12,10,14)(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32)(33,39,40,36)(34,43,38,42)(35,44,37,45)(41,47,46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9947998);