#387154

order	512
nilpotency class	3
isoclinism	isoclinic class of #213564
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#95 ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1268 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)(4,7)(6,8)(17,18)(19,23)(20,25,21,24)(22,26)(27,32,29,30)(28,31);(1,3)(2,4)(5,8)(6,7)(9,10,12,14)(11,15,16,13)(17,19,22,28)(18,23,26,31)(20,29,21,27)(24,32,25,30);(1,4,6,5)(2,7,8,3)(9,11,12,16)(10,13,14,15)(17,20,22,21)(18,24,26,25)(19,27,28,29)(23,30,31,32);(1,5,6,4)(2,7,8,3)(17,21,22,20)(18,25,26,24)(19,27,28,29)(23,30,31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=eg,d^2=ei,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, 294350);