#161104

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	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

822 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)(11,18,20,23)(12,21)(13,16)(14,17)(15,19,24,22)(25,26)(27,35)(28,33)(29,39)(30,34)(31,36)(32,37)(38,40);(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21)(25,27)(26,31)(28,36)(29,38)(30,37)(32,33)(34,40)(35,39);(2,5)(6,8)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)(25,27)(26,32)(28,38)(29,36)(30,37)(31,33)(34,35)(39,40);(25,28,30,29)(26,33,34,39)(27,36,37,38)(31,32,40,35)

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=g,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, 68300);