#512799

order	512
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₁₆	C₂⁴
commutator, socle	C₂²	C₂³×C₁₆
prime radical	C₂²×D₄	C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

388 conjugacy classes of subgroups; only the 240 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 240 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)(3,7)(4,6)(5,8)(9,10,12,16,13,17,22,28,14,18,23,29,24,30,34,38,11,15,19,25,20,26,31,35,21,27,32,36,33,37,39,40);(1,3)(2,5)(4,8)(6,7)(10,15)(16,25)(17,26)(18,27)(28,35)(29,36)(30,37)(38,40);(3,8)(5,7)(10,15)(16,25)(17,26)(18,27)(28,35)(29,36)(30,37)(38,40);(2,6)(5,7)

Polycyclic presentation:

2^9|a^2=f,f^2=g,g^2=h,h^2=i,b^a=be,c^a=ci,d^a=dei,c^b=cei

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419995);