#208597

order	512
nilpotency class	3
isoclinism	isoclinic class of #163983
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

766 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(10,14)(11,17)(12,19)(15,22)(16,21)(20,23)(25,26,29,32,28,31,35,37)(27,30,34,39,33,38,36,40);(9,10,12,14,13,16,19,21)(11,15,18,20,17,23,24,22)(25,27,29,36,28,33,35,34)(26,30,37,39,31,38,32,40);(2,5)(6,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,23)(19,24)(21,22)(26,31)(30,38)(32,37)(39,40);(1,3,4,7)(2,6,5,8)(11,17)(15,23)(18,24)(20,22)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=fh,e^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,e^b=ei,g^b=gi,d^c=dhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 115793);