#6494978

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1422 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,7)(4,5)(6,8)(11,15)(14,16)(17,18,24,38,23,28,22,27)(19,29,37,25,33,42,32,40)(20,34,36,41,21,26,35,48)(30,44,46,47,31,43,45,39);(2,5)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,33)(18,25,28,40)(20,30,21,31)(22,37,24,32)(26,39,34,47)(27,42,38,29)(35,46,36,45)(41,44,48,43);(1,3,4,8)(2,6,5,7)(10,13)(14,16)(17,20,23,21)(18,26,28,34)(19,30,33,31)(22,35,24,36)(25,39,40,47)(27,41,38,48)(29,43,42,44)(32,45,37,46);(10,13)(14,16)(17,21,23,20)(18,26,28,34)(19,31,33,30)(22,36,24,35)(25,39,40,47)(27,41,38,48)(29,43,42,44)(32,46,37,45);(9,11,12,15)(10,14,13,16)(20,21)(26,34)(30,31)(35,36)(39,47)(41,48)(43,44)(45,46)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6402174);