#6535452

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1468 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)(10,13)(14,16)(17,18,23,26)(19,29,31,40)(20,34,37,42)(21,28,22,27)(24,33,39,32)(25,35,41,36)(30,44,45,38)(43,46,48,47);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,31)(18,24,26,39)(20,30,37,45)(21,33,22,32)(25,38,41,44)(27,40,28,29)(34,48,42,43)(35,47,36,46);(10,13)(14,16)(17,20,23,37)(18,25,26,41)(19,30,31,45)(21,36,22,35)(24,38,39,44)(27,34,28,42)(29,43,40,48)(32,47,33,46);(1,3,4,7)(2,6,5,8)(11,15)(14,16)(19,31)(24,39)(29,40)(30,45)(32,33)(38,44)(43,48)(46,47);(9,11,12,15)(10,14,13,16)(18,26)(24,39)(25,41)(27,28)(29,40)(34,42)(38,44)(43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fgh,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6442648);