#237542

order	512
nilpotency class	3
isoclinism	isoclinic class of #237527
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1237 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=i,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,d^b=dh,e^b=eh,g^b=gi,d^c=dhi,e^c=ei,f^c=fhi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144738);