#404633

order	512
nilpotency class	3
isoclinism	isoclinic class of #404632
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

701 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ef,c^2=e,d^2=g,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,f^c=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, 311829);