#134066

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	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

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

normal subgroups only

This group has 858 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=ci,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×#134066
maximal normal subgroup of	#72292877, C₂×#134066
maximal quotient of	C₂×#134066, #72293074

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41262);