#419197

order	512
nilpotency class	3
isoclinism	isoclinic class of #417728
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1823 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)(5,6,12,16,11,17,10,26)(7,18,29,32,20,36,19,13)(8,14,27,34,22,35,21,31)(9,23,28,30,25,15,24,33);(5,7,10,19,11,20,12,29)(6,13,16,18,17,32,26,36)(8,9,21,24,22,25,27,28)(14,33,34,23,35,30,31,15);(2,4)(5,8,11,22)(6,14,17,35)(7,9,20,25)(10,27,12,21)(13,30,32,33)(15,18,23,36)(16,31,26,34)(19,28,29,24);(5,9,11,25)(6,15,17,23)(7,8,20,22)(10,28,12,24)(13,31,32,34)(14,18,35,36)(16,30,26,33)(19,27,29,21)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=i,d^2=i,e^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=de,g^b=gi,e^c=ei,e^d=ei,g^d=gi

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, 326393);