#134109

order	512
nilpotency class	4
isoclinism	isoclinic class of #133828
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#125 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

902 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 902 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,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=ei,f^c=fhi,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72292877, C₂×#134109
maximal normal subgroup of	#72292877, C₂×#134109
maximal quotient of	C₂×#134109, #72293070, #72293072

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41305);