#6472305

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1629 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,g^c=gi,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, 6379501);