#9910668

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	#58254	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4150 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=h,e^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, 9817864);