#6475926

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2288 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,29,33,44)(20,25,21,26)(22,36,38,27)(24,37,41,32)(30,45,31,46)(34,43,35,42)(39,48,40,47);(9,11)(10,13)(12,15)(14,16)(17,19,23,33)(18,24,28,41)(20,30,21,31)(22,37,38,32)(25,39,26,40)(27,44,36,29)(34,48,35,47)(42,46,43,45);(1,3,5,4)(2,6,7,8)(11,15)(13,16)(17,20)(18,25)(19,30)(21,23)(22,34)(24,39)(26,28)(27,42)(29,45)(31,33)(32,47)(35,38)(36,43)(37,48)(40,41)(44,46);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(17,21,23,20)(18,26,28,25)(19,31,33,30)(22,35,38,34)(24,40,41,39)(27,43,36,42)(29,46,44,45)(32,48,37,47);(10,14)(13,16)(20,21)(25,26)(30,31)(34,35)(39,40)(42,43)(45,46)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^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, 6383122);