#129526

order	512
nilpotency class	4
isoclinism	isoclinic class of #12750
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1786
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8310
prime radical	#3584	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

792 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=g,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dgi,f^b=fi,g^b=gi,e^c=ei,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225439, #72225465, #72225466
maximal normal subgroup of	#72225439, #72225465, #72225466

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36722);