#398445

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3062	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1658 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:

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

Polycyclic presentation:

2^9|b^2=ei,c^2=fi,d^2=e,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, 305641);