#146158

order	512
nilpotency class	4
isoclinism	isoclinic class of #146153
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2084
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#461
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

211 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 211 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2,5,6,4,8,3,7)(11,17)(14,22)(18,24)(19,23);(1,3,4,5)(2,6,8,7)(10,14,15,22)(11,18)(12,20)(16,23,21,19)(17,24);(2,7)(3,5)(6,8)(9,10,12,21,13,15,20,16)(11,19,18,14,17,23,24,22)

Polycyclic presentation:

2^9|a^2=e,b^2=d,c^2=dgh,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,f^b=fh,h^b=hi,d^c=di,e^c=eg,f^c=fh,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53354);