#6472868

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470860
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37455	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1696 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,34)(21,25)(22,27)(23,31)(24,35)(26,29)(30,44)(32,39)(33,38)(36,40)(37,47)(41,43)(42,45)(46,48);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,29,32,22,33,21,31)(18,23,25,38,27,39,26,28)(20,30,43,46,36,47,35,45)(24,37,40,48,41,44,34,42);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,20,21,35,22,36,29,43)(18,24,26,34,27,41,25,40)(19,30,31,45,33,47,32,46)(23,37,28,42,39,44,38,48);(11,15)(13,16)(17,21,22,29)(18,25,27,26)(19,31,33,32)(20,35,36,43)(23,38,39,28)(24,40,41,34)(30,45,47,46)(37,48,44,42);(10,14)(13,16)(17,21,22,29)(18,25,27,26)(19,32,33,31)(20,35,36,43)(23,28,39,38)(24,40,41,34)(30,46,47,45)(37,42,44,48)

Polycyclic presentation:

2^9|b^2=gi,c^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,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, 6380064);