#411250

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

661 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)(3,8)(4,6)(5,7)(9,10,15,31,16,19,14,20)(11,21,23,27,17,12,24,26)(13,18,29,33,28,34,30,40)(22,25,37,39,32,35,36,38);(9,11,14,23,16,17,15,24)(10,12,20,26,19,21,31,27)(13,22,29,36,28,32,30,37)(18,25,33,38,34,35,40,39);(1,3,5,4)(2,6,7,8)(9,12)(10,17,19,11)(13,25)(14,26)(15,27)(16,21)(18,32,34,22)(20,24,31,23)(28,35)(29,38)(30,39)(33,37,40,36);(1,4,5,3)(2,6,7,8)(9,13,14,29,16,28,15,30)(10,18,20,33,19,34,31,40)(11,22,23,36,17,32,24,37)(12,25,26,38,21,35,27,39)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318446);