#7162140

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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1526 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)(11,15)(14,16)(17,18,22,26)(19,27,30,38)(20,24,33,41)(21,35,36,25)(23,29,39,31)(28,43,32,47)(34,48,44,42)(37,46,40,45);(2,5)(3,7)(10,13)(11,15)(17,19,21,31,22,30,36,29)(18,23,25,27,26,39,35,38)(20,32,34,46,33,28,44,45)(24,40,42,47,41,37,48,43);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,33)(18,24,26,41)(19,28,30,32)(21,34,36,44)(23,37,39,40)(25,42,35,48)(27,43,38,47)(29,46,31,45);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(19,29)(20,33)(21,36)(23,38)(24,41)(25,35)(27,39)(28,45)(30,31)(32,46)(37,43)(40,47);(9,11,12,15)(10,14,13,16)(19,30)(20,33)(23,39)(24,41)(27,38)(29,31)(34,44)(42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=i,d^2=g,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=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, 7069336);