#7563003

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1711 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)(5,6,8,9)(7,11,12,10)(13,14,18,22)(15,19,26,33)(16,20,27,34)(17,21,29,36)(23,30,38,42)(24,31,25,32)(28,35,40,44)(37,41,39,43);(2,3)(6,9)(10,11)(13,15)(14,19)(16,23)(17,24)(18,26)(20,30)(21,31)(22,33)(25,29)(27,38)(28,37)(32,36)(34,42)(35,41)(39,40)(43,44);(1,2,4,3)(13,16,18,27)(14,20,22,34)(15,23,26,38)(17,28,29,40)(19,30,33,42)(21,35,36,44)(24,37,25,39)(31,41,32,43);(1,3,4,2)(15,24)(17,29)(19,31)(21,36)(23,37)(25,26)(28,40)(30,41)(32,33)(35,44)(38,39)(42,43);(5,7,8,12)(6,10,9,11)(16,27)(20,34)(23,38)(28,40)(30,42)(35,44)(37,39)(41,43)

Polycyclic presentation:

2^9|a^2=fi,c^2=gi,d^2=g,e^2=f,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, 7470199);