#7560700

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	#61178	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1736 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,25)(13,28)(14,19)(15,21)(17,24)(18,30)(20,26)(22,37)(23,33)(27,32)(29,39)(31,35)(34,38)(36,40);(2,5)(6,8)(9,11)(10,16)(12,26)(13,29)(14,23)(15,24)(17,21)(18,32)(19,33)(20,25)(22,31)(27,30)(28,39)(34,36)(35,37)(38,40);(1,3,4,7)(2,6,5,8)(9,12,26,11,15,23,14,24)(10,17,19,33,21,16,20,25)(13,27,40,22,31,38,30,29)(18,34,35,37,36,32,28,39);(9,13,14,30,15,31,26,40)(10,18,20,28,21,36,19,35)(11,22,12,27,24,29,23,38)(16,32,33,37,17,34,25,39);(2,5)(6,8)(9,14,15,26)(10,19,21,20)(11,23,24,12)(13,30,31,40)(16,33,17,25)(18,35,36,28)(22,38,29,27)(32,37,34,39)

Polycyclic presentation:

2^9|c^2=fgi,d^2=g,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,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, 7467896);