#243055

order	512
nilpotency class	3
isoclinism	isoclinic class of #243051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

651 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=e,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,g^a=gh,c^b=cg,d^b=di,e^b=eh,f^b=fh,d^c=dh,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, 150251);