#444412

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2537
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

711 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=g,d^2=f,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,d^b=de,e^b=ei,g^b=gi,e^d=ei,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, 351608);