#476847

order	512
nilpotency class	3
isoclinism	isoclinic class of #476654
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

891 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384043);