#237978

order	512
nilpotency class	3
isoclinism	isoclinic class of #237974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

785 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=i,d^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,f^a=fi,g^a=gh,c^b=cg,d^b=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, 145174);