#106156

order	512
nilpotency class	3
isoclinism	isoclinic class of #8738
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

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

1758 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 1758 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)(11,20,23,34)(12,25,27,17)(13,18)(14,29)(15,19)(16,22,31,24)(21,35)(26,40,28,32)(30,36)(33,38)(37,39);(1,3)(2,5)(4,8)(6,7)(9,11,14,24,15,23,13,22)(10,16,18,20,19,31,29,34)(12,21,28,37,27,38,26,36)(17,30,32,33,25,39,40,35);(9,12)(10,17)(11,21)(13,26)(14,28)(15,27)(16,30)(18,32)(19,25)(20,33)(22,36)(23,38)(24,37)(29,40)(31,39)(34,35)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^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, 13352);