#497699

order	512
nilpotency class	3
isoclinism	isoclinic class of #359901
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2342
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#22554
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

640 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:

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

Polycyclic presentation:

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

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, 404895);