#496668

order	512
nilpotency class	4
isoclinism	isoclinic class of #483439
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂⁵	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

750 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)(11,20,23,34)(12,17)(13,26)(14,29)(15,19)(16,22,31,36)(18,27)(21,33,24,38)(25,32)(28,39)(30,35,40,37);(1,3)(2,5)(4,8)(6,7);(3,8)(5,7)(9,11,13,22,14,24,27,37,15,23,28,36,12,21,25,35)(10,16,18,20,17,30,26,33,19,31,32,34,29,40,39,38);(2,6)(5,7)(9,12,15,14)(10,17,19,29)(11,21,23,24)(13,25,28,27)(16,30,31,40)(18,26,32,39)(20,33,34,38)(22,35,36,37)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265488, #72265504, #72265520, …
maximal normal subgroup of	#72265488, #72265504, #72265520, …
maximal quotient of	#72343790

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403864);