#92914

order	512
nilpotency class	4
isoclinism	isoclinic class of #365
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#106
commutator	C₈	C₄×C₁₆
Frattini, cosocle	C₂×C₈²	C₂²
socle	C₂²	#1277
prime radical	C₂³	#335
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

162 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 90 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)(19,29,31,43)(20,26)(21,36)(22,38)(23,28)(24,30,39,44)(25,34)(27,37)(32,47,45,41)(33,48,46,40)(35,42);(17,19,21,32,37,46,35,44,23,31,34,45,22,33,20,30)(18,24,26,40,38,47,25,29,28,39,42,48,27,41,36,43)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=fh,f^2=h,g^2=i,b^a=bc,c^a=cf,e^a=ef,f^a=fh,d^b=dh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227192, #72227269
maximal normal subgroup of	#72227192, #72227269
maximal quotient of	#72227190

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110);