#162364

order	512
nilpotency class	3
isoclinism	isoclinic class of #162342
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	#2541	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

865 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:

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

Polycyclic presentation:

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

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, 69560);