#146578

order	512
nilpotency class	4
isoclinism	isoclinic class of #146575
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10735
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂³×D₄	C₂³
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1757 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,d^b=dh,e^b=ei,f^b=fh,h^b=hi,d^c=di,f^d=fi,g^d=gi,g^f=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238715, #72238716, #72238723, …
maximal normal subgroup of	#72238715, #72238716, #72238723, …
maximal quotient of	#72240679

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53774);