#7162138

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1504 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)(19,27,31,39)(20,33,35,24)(21,37,28,25)(23,30,40,32)(29,45,34,43)(36,42,44,48)(38,46,41,47);(2,5)(3,7)(10,13)(11,15)(17,19,28,30,22,31,21,32)(18,23,37,39,26,40,25,27)(20,34,44,46,35,29,36,47)(24,41,48,43,33,38,42,45);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,24)(19,29)(21,36)(22,35)(23,38)(25,42)(26,33)(27,43)(28,44)(30,47)(31,34)(32,46)(37,48)(39,45)(40,41);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(19,30)(20,35)(21,28)(23,39)(24,33)(25,37)(27,40)(29,46)(31,32)(34,47)(38,43)(41,45);(9,11,12,15)(10,14,13,16)(19,31)(20,35)(23,40)(24,33)(27,39)(30,32)(36,44)(42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,d^2=gi,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, 7069334);