#6467111

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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	#35210	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2002 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:

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

Polycyclic presentation:

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

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, 6374307);