#475562

order	512
nilpotency class	3
isoclinism	isoclinic class of #460330
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

759 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 382758);