#511330

order	512
nilpotency class	4
isoclinism	isoclinic class of #509117
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

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

normal subgroups only

This group has 378 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)(4,20)(11,32,38,21)(12,28,40,56)(16,41,48,46)(17,33,49,29)(22,23)(24,50)(34,60)(37,57,61,52)(42,62)(45,63)(47,59,64,55)(51,58)(53,54);(1,2,5,25,8,13,6,9)(3,15,14,39,19,44,18,43)(4,21,23,38,20,32,22,11)(7,28,26,12,30,56,27,40)(10,33,35,49,36,29,31,17)(16,45,41,62,48,63,46,42)(24,37,54,52,50,61,53,57)(34,47,51,59,60,64,58,55);(1,3,5,14,8,19,6,18)(2,10,25,35,13,36,9,31)(4,16,23,41,20,48,22,46)(7,29,26,17,30,33,27,49)(11,34,21,51,38,60,32,58)(12,39,56,44,40,43,28,15)(24,55,54,47,50,59,53,64)(37,45,52,62,61,63,57,42);(1,4)(2,11)(3,16)(5,23)(6,22)(7,24)(8,20)(9,32)(10,34)(12,37)(13,38)(14,41)(15,42)(17,47)(18,46)(19,48)(21,25)(26,54)(27,53)(28,57)(29,55)(30,50)(31,58)(33,59)(35,51)(36,60)(39,45)(40,61)(43,63)(44,62)(49,64)(52,56)

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=defi,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418526);