#106160

order	512
nilpotency class	3
isoclinism	isoclinic class of #8738
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#381
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1718 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 1718 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,f^b=fhi,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13356);