#10349017

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	#42059	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4037 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)(17,18,22,20)(19,25,27,23)(21,28,30,24)(26,31,29,32)(33,34)(35,44)(36,40)(37,41)(38,45)(39,43)(42,48)(46,47);(2,3)(5,13)(7,9)(12,16)(17,19,22,27)(18,23,20,25)(21,29,30,26)(24,32,28,31)(35,36)(38,48)(40,44)(42,45);(4,11)(7,9)(12,16)(14,15)(18,20)(23,25)(24,28)(31,32)(33,35,39,36)(34,40,43,44)(37,42,47,45)(38,46,48,41);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,32)(29,30)(33,36,39,35)(34,40,43,44)(37,42,47,45)(38,41,48,46);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,20,22,18)(19,25,27,23)(21,28,30,24)(26,32,29,31)(33,37,39,47)(34,41,43,46)(35,45,36,42)(38,40,48,44);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21,22,30)(18,24,20,28)(19,26,27,29)(23,31,25,32)(33,38,39,48)(34,42,43,45)(35,46,36,41)(37,40,47,44)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=i,d^2=gh,e^2=h,f^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, 10256213);