#494080

order	512
nilpotency class	4
isoclinism	isoclinic class of #483399
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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

646 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)(3,6)(4,5)(7,8)(10,16,18,19)(11,21,23,22)(12,25)(13,27)(14,26)(17,31)(20,35,29,30)(24,38,36,37)(32,34)(39,40);(2,5)(6,8)(9,10,13,19,14,20,27,35,15,18,12,16,26,29,25,30)(11,17,21,33,24,34,37,40,23,32,22,28,36,31,38,39);(1,3,4,7)(2,6,5,8)(9,11,12,22,14,24,25,38,15,23,13,21,26,36,27,37)(10,17,16,28,20,34,30,39,18,32,19,33,29,31,35,40);(10,18)(16,19)(17,32)(20,29)(28,33)(30,35)(31,34)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240839, #72264781, #72264806, …
maximal normal subgroup of	#72240839, #72264781, #72264806, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401276);