#7560256

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	#45412	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1843 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,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, 7467452);