#7166226

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2428	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1735 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2)(3,4)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22,27,32)(23,24,34,29)(25,31,26,30)(28,36,35,33);(7,13)(8,16)(10,17)(11,15)(21,23)(22,28)(24,30)(25,33)(26,36)(27,34)(29,31)(32,35);(2,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23)(22,29)(24,32)(25,36)(26,33)(27,34)(28,31)(30,35);(7,13)(10,17)(14,19)(18,20)(21,24,27,29)(22,23,32,34)(25,35,26,28)(30,33,31,36);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(21,25,27,26)(22,30,32,31)(23,33,34,36)(24,35,29,28)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7073422);