#6472839

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	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2454 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,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, 6380035);