#462524

order	512
nilpotency class	3
isoclinism	isoclinic class of #455397
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2700
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16435
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

715 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 369720);