#6697279

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1570 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)(10,14)(11,12)(17,18,23,28)(19,24,32,41)(20,25,22,27)(21,34,36,26)(29,38,30,39)(31,46,47,40)(33,44,37,42)(35,48,45,43);(9,10)(11,15)(12,16)(13,14)(17,19,23,32)(18,24,28,41)(20,30,22,29)(21,35,36,45)(25,39,27,38)(26,43,34,48)(31,37,47,33)(40,44,46,42);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(19,29)(20,22)(21,36)(24,38)(25,27)(26,34)(30,32)(31,45)(35,47)(39,41)(40,48)(43,46);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(17,20,23,22)(18,25,28,27)(19,30,32,29)(21,33,36,37)(24,39,41,38)(26,42,34,44)(31,45,47,35)(40,48,46,43);(10,14)(15,16)(17,21,23,36)(18,26,28,34)(19,31,32,47)(20,33,22,37)(24,40,41,46)(25,42,27,44)(29,35,30,45)(38,43,39,48)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6604475);