#508804

order	512
nilpotency class	4
isoclinism	isoclinic class of #508669
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3150	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

501 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 501 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,5)(3,6,7,8)(9,10,14,19,16,12,15,20)(11,21,24,27,26,37,33,17)(13,28,30,29,32,18,31,36)(22,23,35,39,34,40,38,25);(2,5)(6,8)(9,11)(10,17)(12,27)(13,23)(14,33)(15,24)(16,26)(18,34)(19,37)(20,21)(22,28)(25,30)(29,38)(31,39)(32,40)(35,36);(3,7)(6,8)(9,12,16,10)(11,22,26,34)(13,29,32,36)(14,19,15,20)(17,23,27,40)(18,30,28,31)(21,25,37,39)(24,38,33,35);(1,3,4,7)(2,6,5,8)(9,13,14,30,16,32,15,31)(10,18,19,36,12,28,20,29)(11,23,33,25,26,40,24,39)(17,34,37,35,27,22,21,38)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416000);