#6441880

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2016 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,23,29,38)(20,24,32,41)(21,25,35,44)(27,36,45,47)(28,37,30,39)(31,40,46,48)(33,42,34,43);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,31)(21,28)(22,29)(24,40)(25,37)(26,38)(27,33)(30,35)(32,46)(34,45)(36,42)(39,44)(41,48)(43,47);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(17,20)(18,24)(19,27)(21,34)(22,32)(23,36)(25,43)(26,41)(28,31)(29,45)(30,46)(33,35)(37,40)(38,47)(39,48)(42,44);(1,4,5,3)(2,6,8,7)(19,28)(21,35)(23,37)(25,44)(27,31)(29,30)(33,34)(36,40)(38,39)(42,43)(45,46)(47,48);(19,29)(20,32)(23,38)(24,41)(28,30)(33,34)(37,39)(42,43)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=f,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349076);