#7515351

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45094	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1973 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(5,6)(7,10)(8,9)(11,12)(14,20)(15,25)(16,29)(17,19)(21,37)(22,39)(26,35)(28,34)(30,36)(32,33)(38,40)(41,43);(1,2)(3,4)(6,9)(10,12)(13,14,18,20)(15,26,28,37)(16,30,33,39)(17,24,19,23)(21,34,35,25)(22,32,36,29)(27,38,31,40)(41,44,43,42);(5,7,8,11)(6,10,9,12)(13,15,18,28)(14,21,20,35)(16,31,33,27)(17,25,19,34)(22,40,36,38)(23,26,24,37)(29,43,32,41)(30,44,39,42);(2,4)(13,16,18,33)(14,22,20,36)(15,27,28,31)(17,29,19,32)(21,38,35,40)(23,30,24,39)(25,41,34,43)(26,42,37,44);(6,9)(10,12)(13,17,18,19)(14,23,20,24)(15,25,28,34)(16,32,33,29)(21,26,35,37)(22,39,36,30)(27,43,31,41)(38,44,40,42)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=i,e^2=i,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,e^b=ei,g^b=gi,h^b=hi,d^c=di,g^c=gi,h^c=hi,e^d=ei,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, 7422547);