#7093446

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1384 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)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,29)(20,33)(21,26)(22,38)(23,28)(24,32)(25,35)(27,30)(31,41)(34,43)(36,42)(37,40)(39,48)(44,47)(45,46);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,19,30,37,23,21,22,32)(18,24,38,26,28,40,27,29)(20,31,44,48,35,34,36,45)(25,39,47,41,33,46,42,43);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,31)(21,34)(22,36)(23,35)(24,39)(26,41)(27,42)(28,33)(29,43)(30,44)(32,45)(37,48)(38,47)(40,46);(11,16)(13,15)(17,21,23,19)(18,26,28,29)(20,34,35,31)(22,32,30,37)(24,27,40,38)(25,41,33,43)(36,45,44,48)(39,42,46,47);(1,4,5,3)(2,7,8,6)(10,14)(13,15)(19,21)(20,35)(24,40)(25,33)(26,29)(32,37)(36,44)(42,47)

Polycyclic presentation:

2^9|b^2=fg,c^2=ghi,d^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^c=ei,e^d=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7000642);