#92824

order	512
nilpotency class	4
isoclinism	isoclinic class of #1355
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#329
commutator	C₄²	C₂×C₁₆
Frattini, cosocle	#1744	C₂²
socle	C₂²	#1270
prime radical	C₂³	#323
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1744 ◁ G
LECS	G ▷ #1744 ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 187 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) = 128

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)(19,27,29,42,30,39,34,48)(20,31,32,46)(21,25)(22,26)(23,28,36,43,38,45,37,44)(24,33,40,47)(35,41);(17,19,22,30)(18,23,26,38)(20,28,32,45)(21,34,35,29)(24,39,40,27)(25,36,41,37)(31,42,46,48)(33,43,47,44)

Polycyclic presentation:

2^9|a^2=d,b^2=h,c^2=f,d^2=g,e^2=h,g^2=i,b^a=bc,c^a=ce,f^a=fh,c^b=cf,d^b=defh,e^b=eh,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, 20);