#8926569

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	C₂×#538	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4064 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833765);