#407779

order	512
nilpotency class	3
isoclinism	isoclinic class of #394464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

625 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)(3,5)(4,6)(7,8)(9,10,14,20,16,21,15,31)(11,22,25,33,26,37,27,17)(12,18,13,19,30,35,29,36)(23,38,39,32,24,34,28,40);(1,3,4,7)(2,5,6,8)(9,11)(10,17)(12,28)(13,24)(14,27)(15,25)(16,26)(18,34)(19,32)(20,37)(21,33)(22,31)(23,29)(30,39)(35,38)(36,40);(9,12,14,13,16,30,15,29)(10,18,20,19,21,35,31,36)(11,23,25,39,26,24,27,28)(17,32,22,34,33,40,37,38);(2,6)(5,8)(9,13,14,30,16,29,15,12)(10,19,20,35,21,36,31,18)(11,24,27,39,26,23,25,28)(17,32,37,38,33,40,22,34)

Polycyclic presentation:

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

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, 314975);