#141255

order	512
nilpotency class	4
isoclinism	isoclinic class of #12430
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#466
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

372 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfghi,c^2=eg,d^2=ghi,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,f^b=fh

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48451);