#10336778

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

4085 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)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,18)(19,24)(20,26)(21,27)(22,28)(23,25)(29,32)(30,31)(34,40)(35,36)(38,48)(43,46);(2,3)(5,13)(7,9)(12,16)(17,19)(18,24)(20,21)(22,25)(23,28)(26,27)(29,30)(31,32)(33,34)(35,41)(36,45)(37,38)(39,40)(42,48)(43,47)(44,46);(4,11)(7,9)(12,16)(14,15)(18,25)(19,28)(21,29)(26,31)(33,35,39,36)(34,41,40,45)(37,46,42,43)(38,44,48,47);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,20)(18,26)(19,29)(21,28)(22,27)(23,30)(24,32)(25,31)(33,36,39,35)(34,41,40,45)(37,46,42,43)(38,47,48,44);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,21,23,29)(18,27,25,32)(19,20,28,30)(22,31,24,26)(33,37,39,42)(34,38,40,48)(35,43,36,46)(41,47,45,44);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,22)(18,19)(20,27)(21,31)(23,24)(25,28)(26,29)(30,32)(33,38)(34,42)(35,44)(36,47)(37,40)(39,48)(41,43)(45,46)

Polycyclic presentation:

2^9|c^2=hi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10243974);