#438260

order	512
nilpotency class	3
isoclinism	isoclinic class of #422131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

839 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=f,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^c=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, 345456);