#92820

order	512
nilpotency class	4
isoclinism	isoclinic class of #1339
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#327
commutator	C₂×C₈	C₂×C₁₆
Frattini, cosocle	C₂×#322	C₂²
socle	C₂²	#1270
prime radical	C₂³	D₄⋊C₈
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×#322 ▷ C₂×C₄² ▷ C₂² ▷ C₁

176 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 54 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 — 2 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=h,c^2=fh,d^2=g,f^2=h,g^2=i,b^a=bc,c^a=ce,e^a=eh,c^b=cf,d^b=def,f^b=fh,g^b=gh,d^c=dh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16);