#443897

order	512
nilpotency class	3
isoclinism	isoclinic class of #428071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

776 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)(5,6,10,13,11,14,17,15)(7,16,19,23,21,32,20,8)(9,24,26,31,27,12,28,30)(18,33,25,22,35,29,34,36);(5,7,17,19,11,21,10,20)(6,8,15,16,14,23,13,32)(9,25,28,35,27,34,26,18)(12,29,31,36,24,33,30,22);(5,8,11,23)(6,7)(9,22,27,36)(10,16,17,32)(12,18)(13,19)(14,21)(15,20)(24,35)(25,30)(26,29,28,33)(31,34);(2,4)(5,9,10,26,11,27,17,28)(6,12,13,30,14,24,15,31)(7,18,19,25,21,35,20,34)(8,22,16,29,23,36,32,33)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 351093);