#7516515

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	#61073	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1935 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,25)(13,31)(14,19)(15,21)(17,29)(18,23)(20,32)(22,34)(24,37)(26,33)(27,35)(28,36)(30,38)(39,40);(2,5)(6,8)(9,11,15,13)(10,16,21,31)(12,26,30,39)(14,23,32,24)(17,34,36,35)(18,20,37,19)(22,28,27,29)(25,33,38,40);(9,12,15,30)(10,17,21,36)(11,22,13,27)(14,28,32,29)(16,33,31,40)(18,35,37,34)(19,25,20,38)(23,26,24,39);(1,3,4,7)(2,6,5,8)(9,13,15,11)(10,18,21,37)(12,27,30,22)(14,24,32,23)(16,19,31,20)(17,35,36,34)(25,40,38,33)(26,28,39,29);(2,5)(6,8)(9,14,15,32)(10,19,21,20)(11,23,13,24)(12,28,30,29)(16,18,31,37)(17,25,36,38)(22,26,27,39)(33,35,40,34)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=fi,e^2=i,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, 7423711);