#152086

order	512
nilpotency class	5
isoclinism	isoclinic class of #152085
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12695
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#127	C₂³
prime radical	#3508	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ #117 ◁ #3508 ◁ G
LECS	G ▷ C₂×#127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

618 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dhi,f^b=fgh,g^b=gi,e^c=ei,f^d=fhi,g^f=gi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59282);