#6443780

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2086 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,h^c=hi,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, 6350976);