#132153

order	512
nilpotency class	4
isoclinism	isoclinic class of #131967
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	#35676	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#124 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1451 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 1451 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268780
maximal normal subgroup of	#72268780
maximal quotient of	#72268782

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 39349);