#236552

order	512
nilpotency class	3
isoclinism	isoclinic class of #236535
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	#60951	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1243 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)(3,7)(4,6)(5,9)(8,13)(10,16)(11,15)(12,14)(17,18)(19,27)(20,24)(21,25)(22,26)(23,29)(28,32)(30,31);(2,6)(3,10)(4,13)(8,9)(11,12)(14,16)(17,19,22,30)(18,23,26,32)(20,29,21,28)(24,31,25,27);(1,3)(2,7)(4,12)(5,11)(6,14)(8,16)(9,15)(10,13)(18,24)(19,28)(20,21)(25,26)(27,31)(29,30);(3,11)(7,15)(10,12)(14,16)(17,20,22,21)(18,24,26,25)(19,28,30,29)(23,27,32,31)

Polycyclic presentation:

2^9|b^2=i,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=cg,d^b=di,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fi,g^c=gh

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, 143748);