#129519

order	512
nilpotency class	4
isoclinism	isoclinic class of #12741
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×#135	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

835 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 835 conjugacy classes of subgroups — too many to draw. Only its 140 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dg,g^b=gi,f^c=fi

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, 36715);