#497032

order	512
nilpotency class	4
isoclinism	isoclinic class of #483987
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3050	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

592 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)(3,7)(4,6)(5,8)(9,10)(11,24)(12,27,31,35)(13,18)(14,22)(15,33)(16,21)(17,23)(19,30,37,28)(20,26)(25,36,39,38)(29,40,32,34);(1,3)(2,5)(4,8)(6,7)(9,11,16,15,17,26,14,13)(10,18,22,20,23,33,21,24)(12,28,29,39,31,30,32,25)(19,35,36,40,37,27,38,34);(3,8)(5,7)(9,12,15,30,16,32,26,39,17,31,13,28,14,29,11,25)(10,19,20,27,22,38,33,40,23,37,24,35,21,36,18,34);(2,6)(5,7)(9,13,16,11,17,15,14,26)(10,20,22,33,23,24,21,18)(12,28,32,25,31,30,29,39)(19,27,38,40,37,35,36,34)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264744, #72264753, #72264874, …
maximal normal subgroup of	#72264744, #72264753, #72264874, …
maximal quotient of	#72343640

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404228);