#428808

order	512
nilpotency class	3
isoclinism	isoclinic class of #424595
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2477
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15390
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1015 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:

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

Polycyclic presentation:

2^9|a^2=ef,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=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, 336004);