#6469777

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
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

1639 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,21,25,22,26,34,40)(19,27,29,36,30,39,33,23)(20,24,32,38,31,37,44,47)(28,41,43,46,42,48,45,35);(9,10)(11,15)(12,13)(14,16)(17,19,22,30)(18,23,26,36)(20,28,31,42)(21,33,34,29)(24,35,37,46)(25,39,40,27)(32,45,44,43)(38,48,47,41);(2,6)(7,8)(10,13)(14,15)(20,31)(24,37)(28,42)(32,44)(35,46)(38,47)(41,48)(43,45);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,20,22,31)(18,24,26,37)(19,28,30,42)(21,32,34,44)(23,35,36,46)(25,38,40,47)(27,41,39,48)(29,43,33,45);(1,4,5,3)(2,8,6,7)(20,31)(24,37)(28,42)(32,44)(35,46)(38,47)(41,48)(43,45)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=gi,e^2=gh,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6376973);