#506270

order	512
nilpotency class	4
isoclinism	isoclinic class of #500850
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3058
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#28143
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

556 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 128 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217287, #72320748, #72321005, …
maximal normal subgroup of	#72217287, #72320748, #72321005, …
maximal quotient of	C₂×#506270

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413466);