#8926573

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	#58416	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 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)(4,6)(5,8)(9,10,15,20)(11,18,13,16)(12,14,19,17)(21,24,23,22)(25,26,31,32)(27,33,40,47)(28,34,43,48)(29,35,44,39)(30,36,46,38)(37,49,53,55)(41,50,45,52)(42,51,56,54);(1,2,4,6)(3,5,8,7)(9,11,15,13)(10,16,20,18)(12,23,19,21)(14,24,17,22)(26,32)(29,44)(33,47)(34,48)(36,38)(41,45)(49,55)(51,54);(1,2)(3,5)(4,6)(7,8)(9,12,15,19)(10,17,20,14)(11,21,13,23)(16,24,18,22)(25,27)(26,33)(28,37)(29,38)(30,35)(31,40)(32,47)(34,49)(36,44)(39,46)(41,51)(42,52)(43,53)(45,54)(48,55)(50,56);(1,3,4,8)(2,5,6,7)(9,13,15,11)(10,18,20,16)(12,23,19,21)(14,24,17,22)(25,28,31,43)(26,34,32,48)(27,37,40,53)(29,45,44,41)(30,42,46,56)(33,49,47,55)(35,52,39,50)(36,51,38,54);(9,14,15,17)(10,19,20,12)(11,22,13,24)(16,23,18,21)(25,29)(26,35)(27,38)(28,41)(30,33)(31,44)(32,39)(34,50)(36,40)(37,51)(42,55)(43,45)(46,47)(48,52)(49,56)(53,54);(9,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,24)(22,23)(25,30)(26,36)(27,39)(28,42)(29,33)(31,46)(32,38)(34,51)(35,40)(37,50)(41,55)(43,56)(44,47)(45,49)(48,54)(52,53)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=h,d^2=g,e^2=h,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, 8833769);