#501025

order	512
nilpotency class	4
isoclinism	isoclinic class of #501024
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24354
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#62636	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

970 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 970 conjugacy classes of subgroups — too many to draw. Only its 160 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:

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

Polycyclic presentation:

2^9|a^2=fi,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,g^b=gh,h^b=hi,e^c=ei,f^c=fi,g^c=gh,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, 408221);