#438261

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

871 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)(5,6,10,15,12,17,11,16)(7,18,20,28,22,8,21,13)(9,14,25,31,27,33,26,32)(19,30,24,36,35,23,34,29);(5,7,11,20,12,22,10,21)(6,13,16,18,17,28,15,8)(9,24,26,35,27,34,25,19)(14,30,32,36,33,23,31,29);(5,8,11,28,12,18,10,13)(6,7,15,20,17,22,16,21)(9,23,26,36,27,30,25,29)(14,19,31,24,33,35,32,34);(2,4)(5,9,10,25,12,27,11,26)(6,14,15,31,17,33,16,32)(7,19,20,24,22,35,21,34)(8,23,13,29,18,30,28,36)

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=fi,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, 345457);