#400685

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23390	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1477 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)(7,8)(9,10)(11,22)(12,27)(13,21)(14,20)(15,19)(16,24)(17,25)(18,30)(23,31)(26,32)(28,35)(29,34)(33,37)(36,39)(38,40);(2,4)(9,11,15,26,14,25,13,24)(10,16,21,17,20,32,19,22)(12,23,30,39,29,38,28,37)(18,31,27,33,35,40,34,36);(2,4)(9,11)(10,17)(12,23)(13,24)(14,25)(15,26)(16,19)(18,33)(20,22)(21,32)(27,40)(28,37)(29,38)(30,39)(31,34)(35,36);(6,8)(9,12,15,30,14,29,13,28)(10,18,21,27,20,35,19,34)(11,23,26,39,25,38,24,37)(16,31,17,33,32,40,22,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 307881);