#469897

order	512
nilpotency class	3
isoclinism	isoclinic class of #459444
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

757 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)(5,6,10,16,11,14,28,15)(7,17,20,30,21,12,27,31)(8,22,24,9,25,32,13,26)(18,35,34,33,23,36,29,19);(5,7,11,21)(6,12,14,17)(8,23,25,18)(9,19,26,33)(10,27,28,20)(13,29,24,34)(15,31,16,30)(22,35,32,36);(5,8,10,24,11,25,28,13)(6,9,15,22,14,26,16,32)(7,18,20,34,21,23,27,29)(12,19,30,36,17,33,31,35);(2,4)(5,9)(6,13,14,24)(7,19)(8,16,25,15)(10,22)(11,26)(12,29,17,34)(18,31,23,30)(20,36)(21,33)(27,35)(28,32)

Polycyclic presentation:

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

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, 377093);