#497558

order	512
nilpotency class	4
isoclinism	isoclinic class of #481415
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3339	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

839 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310206, #72310208, #72310220, …
maximal normal subgroup of	#72310206, #72310208, #72310220, …
maximal quotient of	#72343365, #72343431

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404754);