#7551282

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519083
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61060	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1956 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=g,e^2=fi,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7458478);