#213897

order	512
nilpotency class	3
isoclinism	isoclinic class of #213828
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1652 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 1652 conjugacy classes of subgroups — too many to draw. Only its 206 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,f^b=fh,g^b=gh,g^c=gi,g^d=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121093);