#10023042

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

4109 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)(19,23)(20,21)(22,24)(25,26)(27,32)(28,31)(29,30)(33,34)(35,42)(36,40)(37,45)(38,39)(41,47)(43,44)(46,48);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19,22,27)(18,23,24,32)(20,30,26,28)(21,29,25,31)(33,35,38,40)(34,36,39,42)(37,43,47,48)(41,44,45,46);(2,6)(7,10)(8,12)(14,16)(18,24)(19,27)(20,26)(29,31)(34,39)(36,42)(41,45)(44,46);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,20,22,26)(18,25,24,21)(19,28,27,30)(23,29,32,31)(33,34)(35,42)(36,40)(37,41)(38,39)(43,46)(44,48)(45,47);(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,16)(14,15)(17,19)(18,23)(20,28)(21,31)(22,27)(24,32)(25,29)(26,30)(33,36,38,42)(34,40,39,35)(37,46,47,44)(41,43,45,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21,22,25)(18,26,24,20)(19,29,27,31)(23,28,32,30)(33,37,38,47)(34,41,39,45)(35,43,40,48)(36,44,42,46)

Polycyclic presentation:

2^9|b^2=h,d^2=gh,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, 9930238);