#131613

order	512
nilpotency class	4
isoclinism	isoclinic class of #131224
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×C₄⋊C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 635 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)(12,25)(13,28,14,18)(15,19)(16,22,32,23)(17,26,33,29)(21,30)(24,31)(27,38)(34,35,39,37)(36,40);(1,3)(2,5)(4,6)(7,8)(9,11,15,24)(10,16,19,32)(12,21,27,36)(13,22,14,23)(17,30,33,40)(18,20,28,31)(25,34,38,39)(26,35,29,37);(4,8)(6,7)(9,12)(10,17,19,33)(11,21)(13,29)(14,26)(15,27)(16,30,32,40)(18,25,28,38)(20,34,31,39)(22,37)(23,35)(24,36)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dh,e^b=eg,f^b=fgh,d^c=dgi,f^c=fg,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, 38809);