#9625468

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	#52489	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4115 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 56 points:

(10,11)(15,17)(16,18)(22,23)(25,26,31,32)(27,28,36,33)(29,34,41,45)(30,43,44,35)(37,39,49,46)(38,47,50,40)(42,54,53,48)(51,55,56,52);(1,2)(3,4)(5,6)(7,8)(9,10,14,11)(12,15,19,17)(13,16,21,18)(20,22,24,23)(26,32)(27,36)(29,41)(35,43)(38,50)(39,46)(42,53)(52,55);(2,6)(3,7)(12,19)(15,17)(20,24)(22,23)(25,27,31,36)(26,33,32,28)(29,37,41,49)(30,38,44,50)(34,46,45,39)(35,47,43,40)(42,51,53,56)(48,55,54,52);(3,7)(4,8)(9,11,14,10)(12,15,19,17)(13,16,21,18)(20,23,24,22)(25,28,31,33)(26,27,32,36)(29,39,41,46)(30,40,44,47)(34,37,45,49)(35,38,43,50)(42,52,53,55)(48,51,54,56);(1,3)(2,4)(5,7)(6,8)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,29)(26,34)(27,37)(28,39)(30,42)(31,41)(32,45)(33,46)(35,48)(36,49)(38,51)(40,52)(43,54)(44,53)(47,55)(50,56);(1,4)(2,3)(5,8)(6,7)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,30,31,44)(26,35,32,43)(27,38,36,50)(28,40,33,47)(29,42,41,53)(34,48,45,54)(37,51,49,56)(39,52,46,55)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9532664);