#251817

order	512
nilpotency class	3
isoclinism	isoclinic class of #251809
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2404
prime radical	#18126	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

872 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:

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

Polycyclic presentation:

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

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, 159013);