#161600

order	512
nilpotency class	3
isoclinism	isoclinic class of #161570
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

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 68796);