#511329

order	512
nilpotency class	4
isoclinism	isoclinic class of #509118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3199
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#32460
prime radical	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

380 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=efh,c^2=efh,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=ei,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=hi,e^d=ei,f^d=fi

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, 418525);