#387059

order	512
nilpotency class	3
isoclinism	isoclinic class of #213564
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1498 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,26)(20,24,30,23)(21,25)(22,28)(27,31,29,32);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,19,21,28)(18,22,25,26)(20,29,30,27)(23,32,24,31);(1,4,5,8)(2,6,3,7)(11,16)(13,15)(17,20)(18,23)(19,27)(21,30)(22,31)(24,25)(26,32)(28,29);(2,3)(6,7)(10,14)(11,16)(17,20)(18,24)(19,27)(21,30)(22,32)(23,25)(26,31)(28,29)

Polycyclic presentation:

2^9|a^2=h,b^2=egi,c^2=eg,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,h^c=hi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 294255);