#405279

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×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2322 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)(5,6,10,14)(7,11,17,28)(8,18,20,32)(9,22,25,13)(12,24,30,21)(15,33,19,36)(16,31,23,27)(26,35,29,34);(2,4)(5,7)(6,11)(8,19,20,15)(9,23)(10,17)(12,29,30,26)(13,31)(14,28)(16,25)(18,33,32,36)(21,34,24,35)(22,27);(5,8)(6,12)(7,15)(9,24)(10,20)(11,26)(13,32)(14,30)(16,35)(17,19)(18,22)(21,25)(23,34)(27,36)(28,29)(31,33);(9,25)(13,22)(16,23)(18,32)(21,24)(27,31)(33,36)(34,35)

Polycyclic presentation:

2^9|a^2=i,b^2=h,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, 312475);