#9625403

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3964 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,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, 9532599);