#6374341

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342659
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3666 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,e^b=ei,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6281537);