#419204

order	512
nilpotency class	3
isoclinism	isoclinic class of #417617
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2678
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15268
prime radical	#60387	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1830 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,h^c=hi,e^d=ei,f^d=fi,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326400);