#94348

order	512
nilpotency class	5
isoclinism	isoclinic class of #7422
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1347
commutator	C₂×C₁₆	C₄²
Frattini, cosocle	C₂²×M₃₂	C₂²
prime radical	C₂⁴	SD₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂²×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

225 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 225 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 36 points:

(1,2,3,4)(6,13)(7,17)(8,11)(9,19)(10,24)(14,29)(15,32)(16,31)(18,25)(20,36)(21,22)(23,28)(26,35)(27,30)(33,34);(5,6,8,14,12,16,11,27)(7,18,20,34,22,35,23,32)(9,25,24,33,10,26,19,15)(13,28,29,21,31,36,30,17)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efgh,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=ghi,h^b=hi,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1544);