#510515

order	512
nilpotency class	4
isoclinism	isoclinic class of #510514
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31680
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#525	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

550 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 112 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)(3,14,18,9)(4,19)(5,25)(6,13)(7,12)(8,11)(10,21)(15,46,49,62)(16,33,31,35)(17,45,52,34)(20,55,57,53)(22,54)(23,37)(24,40)(26,59)(27,60,61,42)(28,38)(29,41)(30,39)(32,48,44,50)(36,56,63,51)(43,58)(47,64);(1,3,5,16,8,18,7,31)(2,9,25,35,11,14,12,33)(4,20,21,56,24,57,23,51)(6,27,26,52,30,61,29,17)(10,36,40,55,37,63,19,53)(13,42,59,45,39,60,41,34)(15,47,48,22,49,28,50,58)(32,54,46,38,44,43,62,64);(1,4,8,24)(2,10,11,37)(3,15,18,49)(5,23,7,21)(6,28,30,47)(9,32,14,44)(12,40,25,19)(13,43,39,54)(16,50,31,48)(17,51,52,56)(20,27,57,61)(22,29,58,26)(33,46,35,62)(34,53,45,55)(36,42,63,60)(38,41,64,59);(9,14)(32,44)(33,35)(34,45)(36,63)(42,60)(46,62)(53,55)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417711);