#8926598

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	#2539	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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=g,d^2=h,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, 8833794);