#472247

order	512
nilpotency class	3
isoclinism	isoclinic class of #454681
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
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

774 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:

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

Polycyclic presentation:

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

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, 379443);