#135564

order	512
nilpotency class	4
isoclinism	isoclinic class of #135509
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1819
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 650 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) = 32
transitive degree	μ_t(G) = 32

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267645
maximal normal subgroup of	#72267645
maximal quotient of	#72267647

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42760);