#152470

order	512
nilpotency class	5
isoclinism	isoclinic class of #13673
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2196
commutator	#127	C₂²×C₄
Frattini, cosocle	#532	C₂³
socle	C₂	#12964
prime radical	#3147	C₂²
LCS	G ▷ #127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #532 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

366 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 60 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eh,g^a=gh,e^b=eghi,h^b=hi,d^c=dg,e^c=eh,g^c=gi,e^d=eh,g^d=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, 59666);