#6462666

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1431 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=hi,d^2=f,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6369862);