#6696051

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1541 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)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,27)(20,25)(21,23)(22,26)(24,29)(28,38)(30,43)(31,42)(32,36)(33,41)(34,37)(35,40)(39,45)(44,47)(46,48);(1,3,4,8)(2,5,6,7)(18,23)(19,28)(20,31)(22,34)(24,38)(25,41)(27,40)(29,35)(30,45)(32,33)(36,42)(47,48);(10,13)(14,16)(17,19,21,29)(18,24,23,27)(20,30,32,45)(22,35,34,28)(25,39,36,43)(26,38,37,40)(31,46,33,44)(41,48,42,47);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(18,23)(19,29)(25,36)(26,37)(28,35)(30,45)(41,42)(44,46);(3,8)(5,7)(17,20,21,32)(18,25,23,36)(19,30,29,45)(22,31,34,33)(24,39,27,43)(26,42,37,41)(28,44,35,46)(38,47,40,48)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=gi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

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, 6603247);