#93268

order	512
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₈² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	D₈
commutator	C₈	C₈²
Frattini, cosocle	C₄²×C₈	C₂²
socle, prime radical	C₂³	#340
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 284 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) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fi,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efi,f^a=fi,h^a=hi

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, 464);