#6474210

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470860
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	#24266	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1573 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,33)(21,26)(22,27)(23,37)(24,31)(25,36)(29,43)(30,42)(32,39)(34,40)(35,48)(38,44)(41,46)(45,47);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,29)(21,30)(22,31)(23,32)(25,38)(26,39)(27,28)(33,47)(34,44)(35,45)(36,46)(37,42)(40,41)(43,48);(11,15)(13,16)(17,20,23,36,22,35,21,34)(18,25,26,33,27,40,37,48)(19,29,32,46,31,45,30,44)(24,38,39,47,28,41,42,43);(11,15)(13,16)(17,21,22,23)(18,26,27,37)(19,30,31,32)(20,34,35,36)(24,39,28,42)(25,33,40,48)(29,44,45,46)(38,47,41,43);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(19,31)(24,28)(29,45)(30,32)(38,41)(39,42)(43,47)(44,46)

Polycyclic presentation:

2^9|c^2=g,d^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,e^b=ei

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, 6381406);