#9910651

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4091 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=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, 9817847);