#7615005

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1482 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)(9,10,12,14)(11,15,16,13)(19,27)(20,31)(23,28)(24,39)(30,42)(32,46)(34,40)(38,43);(2,6)(7,8)(11,16)(13,15)(17,18,21,25)(19,28,27,23)(20,32,31,46)(22,36,35,26)(24,40,39,34)(29,44,33,47)(30,38,42,43)(37,45,41,48);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,19,21,27)(18,23,25,28)(20,33,31,29)(22,30,35,42)(24,41,39,37)(26,38,36,43)(32,44,46,47)(34,48,40,45);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,20,22,34,21,31,35,40)(18,24,26,32,25,39,36,46)(19,29,30,45,27,33,42,48)(23,37,38,47,28,41,43,44);(2,6)(7,8)(19,27)(20,31)(23,28)(24,39)(30,42)(32,46)(34,40)(38,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fg,d^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=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, 7522201);