#448326

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2404
prime radical	#23406	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1225 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 355522);