#207197

order	512
nilpotency class	3
isoclinism	isoclinic class of #162486
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

788 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)(3,7)(4,5)(6,8)(10,14)(12,19)(15,20)(16,21)(18,23)(22,24)(25,26,30,33,29,32,28,34)(27,35,36,31,38,40,37,39);(2,5)(6,7)(9,10,12,14,13,16,19,21)(11,15,18,20,17,22,23,24)(25,27,29,38)(26,31,32,39)(28,36,30,37)(33,35,34,40);(1,3,4,8)(2,6,5,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(26,32)(31,39)(33,34)(35,40);(11,17)(15,22)(18,23)(20,24)(25,28,29,30)(26,33,32,34)(27,36,38,37)(31,35,39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=fi,d^2=i,e^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,g^b=gi,d^c=dh

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, 114393);