#6433905

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433057
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	#35328	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1932 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)(17,18,22,26)(19,28,21,24)(20,31,33,47)(23,37,36,27)(25,40,41,34)(29,44,32,46)(30,39,35,43)(38,45,42,48);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,21)(18,24,26,28)(20,29,33,32)(23,30,36,35)(25,38,41,42)(27,39,37,43)(31,46,47,44)(34,45,40,48);(11,15)(13,16)(17,20,23,34,22,33,36,40)(18,25,27,31,26,41,37,47)(19,29,30,45,21,32,35,48)(24,38,39,46,28,42,43,44);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,21,22,19)(18,24,26,28)(20,32,33,29)(23,35,36,30)(25,38,41,42)(27,39,37,43)(31,46,47,44)(34,48,40,45);(19,21)(20,33)(24,28)(25,41)(30,35)(31,47)(34,40)(39,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,e^b=ei,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, 6341101);