#136012

order	512
nilpotency class	4
isoclinism	isoclinic class of #136011
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1853
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

439 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 439 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 64

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

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

Polycyclic presentation:

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

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, 43208);