#475495

order	512
nilpotency class	3
isoclinism	isoclinic class of #469927
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2605
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17053
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

617 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 617 conjugacy classes of subgroups — too many to draw. Only its 156 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:

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

Polycyclic presentation:

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

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, 382691);