#123629

order	512
nilpotency class	3
isoclinism	isoclinic class of #2048
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#394
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

2438 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,e^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,h^a=hi,c^b=cf,d^b=di,f^b=fi,g^b=gi,e^c=ei,g^c=gi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30825);