#152084

order	512
nilpotency class	5
isoclinism	isoclinic class of #149794
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2126
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#532	C₂³
socle	C₂	#12695
prime radical	#60673	C₂
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₄² ◁ C₂×C₄² ◁ #3444 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 633 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)(3,13,14,7)(4,11,19,9)(5,10)(6,12)(8,18,27,32)(15,21,29,22)(16,23,17,24)(20,26)(25,31,28,30);(1,3)(2,7)(4,17)(5,15)(6,16)(8,25)(9,23)(10,22)(11,13)(12,24)(14,19)(18,31)(20,30)(21,27)(26,28)(29,32);(1,4,6,19)(2,8)(3,14,16,17)(5,18,20,32)(7,21)(9,26)(10,11)(12,27)(13,28)(15,29,30,31)(22,23)(24,25)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59280);