#6462670

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

1417 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)(17,18,24,37,23,27,22,28)(19,29,21,25,32,44,31,40)(20,26,38,47,35,43,36,42)(30,41,34,39,45,48,33,46);(2,5)(6,8)(17,19,23,32)(18,25,27,40)(20,33,35,34)(21,24,31,22)(26,41,43,48)(28,44,37,29)(30,38,45,36)(39,47,46,42);(10,13)(14,16)(17,20,24,38,23,35,22,36)(18,26,37,47,27,43,28,42)(19,30,21,34,32,45,31,33)(25,39,44,48,40,46,29,41);(1,3,4,7)(2,6,5,8)(17,21,22,19,23,31,24,32)(18,25,28,29,27,40,37,44)(20,34,36,30,35,33,38,45)(26,39,42,41,43,46,47,48);(9,11,12,15)(10,14,13,16)(18,27)(20,35)(25,40)(28,37)(29,44)(30,45)(33,34)(36,38)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,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, 6369866);