#496305

order	512
nilpotency class	3
isoclinism	isoclinic class of #382440
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2309
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#21790
prime radical	#3449	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

927 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,6,13,7,12,27,37,8,11,26,36,5,24,25,52)(3,14,18,40,19,30,47,56,17,39,46,62,16,9,28,31)(4,10,22,34,23,33,51,58,20,35,48,59,21,49,50,64)(15,41,44,63,45,29,54,55,42,38,60,61,43,32,53,57);(1,3,5,16,8,17,7,19)(2,9,12,14,11,30,24,39)(4,15,21,43,20,42,23,45)(6,18,25,28,26,46,27,47)(10,32,33,41,35,29,49,38)(13,31,37,40,36,56,52,62)(22,44,50,53,48,60,51,54)(34,57,58,63,59,55,64,61);(1,4)(2,10)(3,15)(5,21)(6,22)(7,23)(8,20)(9,29)(11,35)(12,33)(13,34)(14,38)(16,43)(17,42)(18,44)(19,45)(24,49)(25,50)(26,48)(27,51)(28,53)(30,32)(31,55)(36,59)(37,58)(39,41)(40,61)(46,60)(47,54)(52,64)(56,57)(62,63);(2,11)(4,20)(6,26)(9,30)(12,24)(14,39)(15,42)(18,46)(21,23)(25,27)(28,47)(34,59)(43,45)(55,57)(58,64)(61,63)

Polycyclic presentation:

2^9|a^2=g,b^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=ceh,f^b=fi,d^c=di,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403501);