#6450712

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1540 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6357908);