#238296

order	512
nilpotency class	3
isoclinism	isoclinic class of #238291
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1020 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 145492);