#405278

order	512
nilpotency class	3
isoclinism	isoclinic class of #394275
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#454	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2232 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=h,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,f^b=fi,g^b=gi,e^c=ei,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312474);