#10038428

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

3961 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,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, 9945624);