#10040511

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3743 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)(3,7,9,10)(5,12,13,8)(11,16,15,14)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,31)(30,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,46)(44,48)(45,47);(2,4)(3,9)(8,12)(11,15)(19,26)(20,29)(23,25)(24,28)(34,39)(40,46)(41,47)(42,48);(2,4)(3,9)(8,12)(11,15)(17,19,22,26)(18,23,21,25)(20,30,29,27)(24,32,28,31)(33,35,38,43)(34,40,39,46)(36,44,45,37)(41,48,47,42);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,20,22,29)(18,24,21,28)(19,27,26,30)(23,31,25,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,46)(44,48)(45,47);(1,4,6,2)(3,10,9,7)(5,8,13,12)(11,14,15,16)(17,19)(18,25)(20,27)(21,23)(22,26)(24,32)(28,31)(29,30)(33,36,38,45)(34,41,39,47)(35,37,43,44)(40,42,46,48);(1,5,6,13)(2,8,4,12)(3,11,9,15)(7,14,10,16)(17,21,22,18)(19,25,26,23)(20,28,29,24)(27,32,30,31)(33,37)(34,42)(35,36)(38,44)(39,48)(40,41)(43,45)(46,47)

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=i,e^2=g,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, 9947707);