#92945

order	512
nilpotency class	4
isoclinism	isoclinic class of #7221
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#340
commutator	C₂×C₈	C₄×C₈
Frattini, cosocle	C₂×C₈²	C₂²
socle	C₂²	#1279
prime radical	C₂³	Q₈⋊C₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

160 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fi,d^2=g,e^2=fg,f^2=h,g^2=i,b^a=bc,c^a=cf,e^a=efhi,f^a=fh,c^b=ch,d^b=dhi

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, 141);