#505675

order	512
nilpotency class	4
isoclinism	isoclinic class of #500676
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

577 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 577 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412871);