#6445469

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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 ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3066 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)(5,6,8,10)(7,11,12,9)(13,14,18,24)(15,26,30,37)(16,22,34,21)(17,25,19,23)(20,28,38,29)(27,42,44,39)(31,36,43,35)(32,40,33,41);(2,4)(13,15)(14,20)(16,31)(17,28)(18,30)(19,29)(21,39)(22,42)(23,37)(24,38)(25,26)(27,32)(33,44)(34,43)(35,40)(36,41);(5,7,8,12)(6,9,10,11)(13,16)(14,21)(15,27)(17,32)(18,34)(19,33)(20,35)(22,24)(23,40)(25,41)(26,39)(28,43)(29,31)(30,44)(36,38)(37,42);(13,16)(14,22)(15,27)(17,32)(18,34)(19,33)(20,36)(21,24)(23,41)(25,40)(26,42)(28,43)(29,31)(30,44)(35,38)(37,39);(13,17,18,19)(14,23,24,25)(15,28,30,29)(16,32,34,33)(20,37,38,26)(21,40,22,41)(27,43,44,31)(35,42,36,39)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352665);