#6480322

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1570 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)(11,12)(14,16)(17,18,22,26)(19,28,31,43)(20,25,33,24)(21,35,37,27)(23,36,40,30)(29,45,32,44)(34,41,48,42)(38,46,39,47);(9,10)(11,16)(12,14)(13,15)(17,19,22,31)(18,23,26,40)(20,32,33,29)(21,36,37,30)(24,39,25,38)(27,43,35,28)(34,46,48,47)(41,44,42,45);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,20,22,33)(18,24,26,25)(19,29,31,32)(21,34,37,48)(23,38,40,39)(27,41,35,42)(28,44,43,45)(30,46,36,47);(10,15)(14,16)(17,20)(18,25)(19,29)(21,34)(22,33)(23,39)(24,26)(27,42)(28,45)(30,46)(31,32)(35,41)(36,47)(37,48)(38,40)(43,44);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,14,15,16)(18,26)(20,33)(23,40)(27,35)(28,43)(29,32)(34,48)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,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, 6387518);