#7519194

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45134	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2636 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=i,e^2=fi,g^2=i,h^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,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7426390);