#7597049

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563337
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	#61079	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1992 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)(11,23)(12,19)(13,32)(14,21)(15,20)(16,22)(17,26)(18,30)(24,33)(25,34)(27,36)(28,35)(29,39)(31,38)(37,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,26,16,28,15,27)(10,17,20,23,22,36,21,35)(12,29,30,24,31,40,13,25)(18,37,19,33,32,39,38,34);(2,5)(6,7)(9,12,15,13,16,31,14,30)(10,18,21,38,22,32,20,19)(11,24,26,40,28,25,27,29)(17,33,23,39,36,34,35,37);(1,3,4,8)(2,6,5,7)(9,12,14,30,16,31,15,13)(10,18,20,19,22,32,21,38)(11,25,27,40,28,24,26,29)(17,34,35,39,36,33,23,37);(9,13,15,31,16,30,14,12)(10,19,21,18,22,38,20,32)(11,25,27,40,28,24,26,29)(17,33,35,37,36,34,23,39)

Polycyclic presentation:

2^9|b^2=f,c^2=h,d^2=g,e^2=fi,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=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, 7504245);