#237412

order	512
nilpotency class	3
isoclinism	isoclinic class of #237395
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	#60916	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1234 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144608);