#161127

order	512
nilpotency class	3
isoclinism	isoclinic class of #160892
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂×#127	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

874 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)(9,10)(11,19,21,24)(12,17)(13,18)(14,16)(15,20,23,22)(25,26)(27,34)(28,30)(29,33)(31,35)(32,37)(36,39)(38,40);(9,11,14,21)(10,15,16,23)(12,20,13,22)(17,19,18,24)(26,30)(28,37)(31,38)(32,33)(34,39)(36,40);(2,5)(6,8)(10,16)(15,23)(17,18)(19,24)(25,27,29,35)(26,31,33,34)(28,36,37,40)(30,38,32,39);(9,12,14,13)(10,17,16,18)(11,20,21,22)(15,19,23,24)(27,35)(31,34)(36,40)(38,39)

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=g,d^2=gh,e^2=h,f^2=i,b^a=be,c^a=chi,e^a=eg,e^b=eh,f^b=fgh,d^c=dg

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 68323);