#471373

order	512
nilpotency class	3
isoclinism	isoclinic class of #460970
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2759
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16791
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

529 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)(3,15,20,34,19,9,18,33)(4,21,24,10,25,41,30,37)(5,26,27,39,29,40,28,11)(16,43,50,31,22,52,45,35)(17,46,49,42,48,56,47,32)(23,38,54,51,55,36,57,60)(44,58,63,64,53,59,62,61);(1,3)(2,9)(4,22,25,16)(5,17)(6,18)(7,20)(8,19)(10,35,37,31)(11,32)(12,15)(13,33)(14,34)(21,43,41,52)(23,53,55,44)(24,50,30,45)(26,56)(27,47)(28,49)(29,48)(36,59,38,58)(39,42)(40,46)(51,61,60,64)(54,63,57,62);(1,4,6,24,8,25,7,30)(2,10,13,21,12,37,14,41)(3,16,18,45,19,22,20,50)(5,23,27,54,29,55,28,57)(9,31,33,52,15,35,34,43)(11,38,40,60,39,36,26,51)(17,44,47,62,48,53,49,63)(32,59,46,61,42,58,56,64);(1,5,6,27,8,29,7,28)(2,11,13,40,12,39,14,26)(3,17,18,47,19,48,20,49)(4,23,24,54,25,55,30,57)(9,32,33,46,15,42,34,56)(10,36,21,51,37,38,41,60)(16,44,45,62,22,53,50,63)(31,58,52,64,35,59,43,61)

Polycyclic presentation:

2^9|a^2=efg,b^2=fh,c^2=f,d^2=f,f^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,g^b=gi,d^c=de,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, 378569);