#497059

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

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	C₂²×D₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

557 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)(9,10,15,20,16,22,13,17)(11,23,24,38,27,37,31,34)(12,28,30,33,29,21,14,19)(18,25,32,26,36,39,35,40);(1,2)(3,6)(4,5)(7,8)(10,17,22,20)(11,24,27,31)(12,29)(14,30)(19,28,33,21)(23,37)(25,40,39,26)(34,38);(2,5)(6,8)(9,11,14,26,13,24,29,39,16,27,30,40,15,31,12,25)(10,18,21,23,20,35,19,34,22,36,28,37,17,32,33,38);(1,3,4,7)(2,6,5,8)(9,12,15,30,16,29,13,14)(10,19,17,21,22,33,20,28)(11,25,31,40,27,39,24,26)(18,34,32,23,36,38,35,37)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=f,d^2=ei,e^2=i,f^2=h,g^2=i,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	#72264905, #72264975, #72265287, …
maximal normal subgroup of	#72264905, #72264975, #72265287, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404255);