#6451701

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437415
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3455	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3898 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:

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

Polycyclic presentation:

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

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, 6358897);