#6470388

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470374
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	#34632	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2861 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)(15,24)(16,21)(17,22)(18,32)(19,23)(20,28)(25,37)(26,34)(27,38)(29,35)(30,36)(31,42)(33,40)(39,43)(41,44);(5,7)(6,9)(8,11)(10,12)(13,15,18,28,19,27,17,26)(14,20,22,24,23,34,32,38)(16,25,31,40,30,41,29,39)(21,33,35,37,36,43,42,44);(2,4)(7,11)(9,12)(13,16)(14,21)(15,25)(17,29)(18,31)(19,30)(20,33)(22,35)(23,36)(24,37)(26,39)(27,41)(28,40)(32,42)(34,43)(38,44);(7,11)(9,12)(13,17,19,18)(14,22,23,32)(15,26,27,28)(16,29,30,31)(20,24,34,38)(21,35,36,42)(25,39,41,40)(33,37,43,44);(6,10)(9,12)(15,27)(16,30)(20,34)(21,36)(24,38)(26,28)(29,31)(35,42)

Polycyclic presentation:

2^9|b^2=f,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^b=ei,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6377584);