#420479

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1706 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,c^2=ei,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=gi

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, 327675);