#7516779

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×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2029 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=fi,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,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, 7423975);