#123645

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	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

2442 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)(11,16,22,32)(12,24)(13,20)(14,19)(15,18)(17,26)(21,36,25,39)(23,35,29,31)(27,33)(28,34)(30,38,40,37);(2,4)(9,11,14,23,15,22,13,29)(10,16,20,31,18,32,19,35)(12,25,27,38,28,21,26,37)(17,30,34,36,33,40,24,39);(6,8)(9,12,15,28)(10,17,18,33)(11,21,22,25)(13,27,14,26)(16,30,32,40)(19,34,20,24)(23,38,29,37)(31,39,35,36)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,h^a=hi,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=gi,e^c=ei,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, 30841);