#207087

order	512
nilpotency class	3
isoclinism	isoclinic class of #161854
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

786 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)(9,10,13,16,14,17,12,22)(11,18,21,23,19,24,20,15)(26,31,33,37)(32,38,40,39);(2,5)(6,7)(9,11,14,19)(10,15)(12,20,13,21)(16,18)(17,23)(22,24)(25,26,29,37,30,33,28,31)(27,32,34,39,35,40,36,38);(1,3,4,8)(2,6,5,7)(11,19)(15,23)(18,24)(20,21)(25,27)(26,32)(28,36)(29,34)(30,35)(31,38)(33,40)(37,39);(25,28,30,29)(26,31,33,37)(27,34,35,36)(32,39,40,38)

Polycyclic presentation:

2^9|a^2=g,b^2=efgh,c^2=f,d^2=i,e^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,c^b=ch,e^b=ei,g^b=gi,d^c=di

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, 114283);