#9935817

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

4102 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)(3,8,11,7)(5,9,14,12)(10,16,13,15)(18,22)(20,28)(23,25)(26,32)(33,34)(35,42)(36,39)(37,45)(38,41)(40,47)(43,44)(46,48);(1,3,6,11)(2,7,4,8)(5,10,14,13)(9,15,12,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35)(34,36)(37,43)(38,39)(40,44)(41,42)(45,46)(47,48);(17,18,21,22)(19,23,27,25)(20,24,28,29)(26,30,32,31)(33,34,38,41)(35,36,39,42)(37,40,47,45)(43,44,48,46);(1,4,6,2)(3,7,11,8)(5,12,14,9)(10,15,13,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,46)(44,48)(45,47);(1,3)(2,8)(4,7)(5,13)(6,11)(9,15)(10,14)(12,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,36,38,42)(34,39,41,35)(37,46,47,44)(40,43,45,48);(1,5)(2,9)(3,10)(4,12)(6,14)(7,15)(8,16)(11,13)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,37,38,47)(34,40,41,45)(35,43,39,48)(36,44,42,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=gh,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=eghi,d^c=dh,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, 9843013);