#139936

order	512
nilpotency class	4
isoclinism	isoclinic class of #139718
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
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

632 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238678, #72238680, #72238682
maximal normal subgroup of	#72238678, #72238680, #72238682

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47132);