#163754

order	512
nilpotency class	3
isoclinism	isoclinic class of #163726
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	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

837 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 70950);