#496732

order	512
nilpotency class	4
isoclinism	isoclinic class of #483449
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#106
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

536 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240839, #72265458, #72265476, …
maximal normal subgroup of	#72240839, #72265458, #72265476, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403928);