#512808

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
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	C₂³	C₂²×C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

312 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 242 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,21,27,22,28,33,42,23,24,34,36,35,37,45,46,20,26,31,40,32,41,44,48,19,25,29,38,30,39,43,47);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,20,23)(18,24,26,25)(21,29,31,34)(22,30,32,35)(27,36,40,38)(28,37,41,39)(33,43,44,45)(42,46,48,47);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19,20,23)(18,25,26,24)(21,29,31,34)(22,30,32,35)(27,38,40,36)(28,39,41,37)(33,43,44,45)(42,47,48,46);(2,8)(6,9)(7,16)(14,15)(18,26)(24,25)(27,40)(28,41)(36,38)(37,39)(42,48)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420004);