#107128

order	512
nilpotency class	3
isoclinism	isoclinic class of #107114
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1698 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1698 conjugacy classes of subgroups — too many to draw. Only its 202 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)(9,10,13,18,15,19,14,28)(11,20,23,34,22,16,24,30)(12,17,26,32,25,31,27,38)(21,29,36,39,35,33,37,40);(1,3)(2,5)(4,8)(6,7)(9,11,14,24,15,22,13,23)(10,16,18,30,19,20,28,34)(12,21,27,37,25,35,26,36)(17,29,32,39,31,33,38,40);(9,12)(10,17)(11,21)(13,26)(14,27)(15,25)(16,29)(18,32)(19,31)(20,33)(22,35)(23,36)(24,37)(28,38)(30,39)(34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14324);