#151832

order	512
nilpotency class	5
isoclinism	isoclinic class of #151827
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12574
commutator	C₂×D₄	C₂×C₄²
Frattini, cosocle	#532	C₂³
prime radical	#3498	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #3498 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

570 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 120 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,d^a=dg,g^a=gh,h^a=hi,c^b=ch,e^b=egi,g^b=gi,h^b=hi,d^c=di,f^c=fi,e^d=ehi,g^d=gi,g^e=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59028);