#237990

order	512
nilpotency class	3
isoclinism	isoclinic class of #237983
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	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

777 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:

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

Polycyclic presentation:

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

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, 145186);