#6450054

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂×#3455	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3052 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6357250);