#213898

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	#35676	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1716 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 1716 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) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=hi,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, 121094);