#6448896

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2284 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)(15,25)(16,29)(17,22)(18,34)(19,24)(20,23)(21,28)(26,36)(27,38)(30,40)(31,41)(32,44)(33,42)(35,39)(37,43);(2,4)(13,15)(14,21)(16,30)(17,32)(18,27)(19,28)(20,35)(22,40)(23,38)(24,25)(26,31)(29,44)(33,43)(34,39)(36,42)(37,41);(6,9)(10,12)(13,16,19,17)(14,22,24,29)(15,26,28,43)(18,31,20,33)(21,36,25,37)(23,41,34,42)(27,32,35,30)(38,40,39,44);(5,7,8,11)(6,10,9,12)(13,17,19,16)(14,22,24,29)(15,26,28,43)(18,33,20,31)(21,37,25,36)(23,41,34,42)(27,32,35,30)(38,44,39,40);(13,18,19,20)(14,23,24,34)(15,27,28,35)(16,31,17,33)(21,38,25,39)(22,41,29,42)(26,32,43,30)(36,40,37,44)

Polycyclic presentation:

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

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, 6356092);