#123601

order	512
nilpotency class	3
isoclinism	isoclinic class of #9091
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

2494 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)(9,10,14,20,15,18,13,19)(11,16,23,30,22,31,28,34)(12,24,26,17,27,33,25,32)(21,35,37,39,36,40,38,29);(2,4)(5,6)(7,8)(9,11)(10,16,18,31)(12,21)(13,23)(14,28)(15,22)(17,29,32,39)(19,30,20,34)(24,35,33,40)(25,37)(26,38)(27,36);(6,8)(9,12,15,27)(10,17,18,32)(11,21,22,36)(13,26,14,25)(16,29,31,39)(19,33,20,24)(23,38,28,37)(30,40,34,35)

Polycyclic presentation:

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

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, 30797);