#7086245

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1411 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:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=hi,d^2=fi,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,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, 6993441);