#407052

order	512
nilpotency class	3
isoclinism	isoclinic class of #407051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14946
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1138 conjugacy classes of subgroups; only the 224 normal subgroups are shown.

normal subgroups only

This group has 1138 conjugacy classes of subgroups — too many to draw. Only its 224 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 314248);