#407035

order	512
nilpotency class	3
isoclinism	isoclinic class of #394590
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1075 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=efh,c^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,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, 314231);