#9481513

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	#53977	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4000 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,d^2=gi,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9388709);