#475498

order	512
nilpotency class	3
isoclinism	isoclinic class of #469927
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2605
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17056
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 541 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=gh,c^2=ef,d^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,f^b=fi,d^c=de,g^c=gi,h^c=hi,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, 382694);