#427316

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
prime radical	#23399	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1343 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 334512);