#131547

order	512
nilpotency class	4
isoclinism	isoclinic class of #131176
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂×C₄⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	#392	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

562 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=h,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=eg,f^b=fgh,d^c=dgi,e^c=ei,f^c=fgi,e^d=ei,f^d=fi,f^e=fi

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, 38743);