#6433252

order	512
nilpotency class	3
isoclinism	isoclinic class of #29736
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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×#3054	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2132 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6340448);