#427710

order	512
nilpotency class	3
isoclinism	isoclinic class of #418530
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2676
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15372
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

928 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:

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

Polycyclic presentation:

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

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, 334906);