#7620366

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

1344 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)(4,8)(5,6)(10,14)(11,12)(17,18,23,28)(19,24,31,38)(20,25,33,40)(21,26,30,37)(22,27,35,42)(29,36,43,46)(32,39,44,47)(34,41,45,48);(2,6)(7,8)(9,10)(11,16)(12,15)(13,14)(17,19,23,31)(18,24,28,38)(20,29,33,43)(21,35,30,22)(25,36,40,46)(26,42,37,27)(32,45,44,34)(39,48,47,41);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,20,23,33)(18,25,28,40)(19,29,31,43)(21,32,30,44)(22,34,35,45)(24,36,38,46)(26,39,37,47)(27,41,42,48);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,15,14,16)(17,21,22,19,23,30,35,31)(18,26,27,24,28,37,42,38)(20,32,34,29,33,44,45,43)(25,39,41,36,40,47,48,46);(2,6)(7,8)(9,12,13,11)(10,15,14,16)(20,33)(25,40)(29,43)(32,44)(34,45)(36,46)(39,47)(41,48)

Polycyclic presentation:

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

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, 7527562);