#136121

order	512
nilpotency class	4
isoclinism	isoclinic class of #136008
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1853
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#106
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

291 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 291 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,d^2=g,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dg,e^b=eh,f^b=fi,g^b=gi,d^c=dh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285149, #72285170, #72285235
maximal normal subgroup of	#72285149, #72285170, #72285235
maximal quotient of	#72287603, #72287609, #72287610, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43317);