#139755

order	512
nilpotency class	4
isoclinism	isoclinic class of #139755
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9495
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	#23904	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

743 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 743 conjugacy classes of subgroups — too many to draw. Only its 66 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)(3,11,14,5)(4,15)(6,20,22,9)(7,10)(8,17,24,29)(12,27)(13,26)(16,28,25,30)(18,31)(19,21)(23,32);(1,3,11,10)(2,7,14,5)(4,12,25,24)(6,21,19,20)(8,18,23,16)(9,22,13,26)(15,29,28,27)(17,30,32,31);(1,4)(2,8)(3,12)(5,16)(6,17)(7,18)(9,15)(10,24)(11,25)(13,28)(14,23)(19,32)(20,31)(21,30)(22,29)(26,27)

Polycyclic presentation:

2^9|a^2=f,b^2=df,d^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,d^b=dgi,e^b=eh,f^b=fgi,g^b=gi,h^b=hi,d^c=dh,f^c=fh,g^c=gi,f^d=fi,f^e=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240500, #72258402
maximal normal subgroup of	#72240500, #72258402
maximal quotient of	#72266219, #72266296

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46951);