#443358

order	512
nilpotency class	3
isoclinism	isoclinic class of #429607
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2595
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15757
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

674 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=efg,c^2=ef,d^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=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, 350554);