#7520545

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514148
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	#30551	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1751 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7427741);