#397261

order	512
nilpotency class	3
isoclinism	isoclinic class of #397260
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14748
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

1163 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 1163 conjugacy classes of subgroups — too many to draw. Only its 232 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:

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

Polycyclic presentation:

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

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, 304457);