#130576

order	512
nilpotency class	4
isoclinism	isoclinic class of #130512
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2356 ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

511 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 511 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37772);