#500267

order	512
nilpotency class	4
isoclinism	isoclinic class of #500262
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24058
commutator, Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#3499	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 811 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:

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

Polycyclic presentation:

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

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, 407463);