#7614274

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563333
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	#61211	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1749 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)(4,8)(5,6)(9,10)(11,23)(12,26)(13,17)(14,22)(15,21)(16,20)(18,30)(19,32)(24,38)(25,34)(27,39)(28,37)(29,36)(31,35)(33,40);(2,6)(7,8)(9,11,15,13)(10,17,21,23)(12,27,29,40)(14,25,16,32)(18,35,37,38)(19,20,34,22)(24,28,31,30)(26,33,36,39);(1,3,5,4)(2,7,6,8)(9,12,16,30,15,29,14,28)(10,18,22,26,21,37,20,36)(11,24,25,27,13,31,32,40)(17,33,34,35,23,39,19,38);(9,13,15,11)(10,19)(12,24,29,31)(14,25,16,32)(17,22)(18,35)(20,23)(21,34)(26,39)(27,28,40,30)(33,36)(37,38);(1,4,5,3)(2,7,6,8)(9,14,15,16)(10,20,21,22)(11,25,13,32)(12,28,29,30)(17,34,23,19)(18,36,37,26)(24,27,31,40)(33,35,39,38)

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=hi,e^2=f,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,d^c=di,e^d=ei

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, 7521470);