#348063

order	512
nilpotency class	3
isoclinism	isoclinic class of #348058
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

856 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:

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

Polycyclic presentation:

2^9|b^2=eh,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,f^c=fi,g^c=gi,f^d=fi,g^d=gh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255259);