#7517009

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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2002 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)(13,14,19,24)(15,20,28,36)(16,29,31,38)(17,23,18,22)(21,33,39,32)(25,37,42,44)(26,35,27,40)(30,34,41,43);(1,2)(3,4)(6,9)(10,12)(13,15,19,28)(14,20,24,36)(16,30,31,41)(17,27,18,26)(21,37,39,44)(22,35,23,40)(25,33,42,32)(29,34,38,43);(13,16,19,31)(14,21,24,39)(15,25,28,42)(17,32,18,33)(20,34,36,43)(22,29,23,38)(26,30,27,41)(35,44,40,37);(2,4)(5,7,8,11)(6,10,9,12)(14,22)(15,26)(16,31)(17,18)(21,38)(23,24)(25,41)(27,28)(29,39)(30,42)(34,43)(35,40);(6,9)(10,12)(13,17,19,18)(14,23,24,22)(15,26,28,27)(16,32,31,33)(20,35,36,40)(21,38,39,29)(25,30,42,41)(34,44,43,37)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=f,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, 7424205);