#9625472

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	#57020	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4119 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,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, 9532668);