#131591

order	512
nilpotency class	4
isoclinism	isoclinic class of #131264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂×C₂²⋊C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	#2617	C₂²
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 708 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=ghi,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dh,e^b=egi,f^b=fgh,d^c=dg,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, 38787);