#138016

order	512
nilpotency class	4
isoclinism	isoclinic class of #137984
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

413 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 413 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=g,d^2=ghi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=ei,f^b=fh,d^c=di,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45212);