#396374

order	512
nilpotency class	3
isoclinism	isoclinic class of #396373
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1383 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)(5,6,10,16,11,17,29,15)(7,18,21,12,23,31,22,32)(8,9,26,27,14,28,25,13)(19,35,30,33,36,20,24,34);(5,7,10,21,11,23,29,22)(6,12,15,18,17,32,16,31)(8,24,25,36,14,30,26,19)(9,20,27,34,28,35,13,33);(5,8,11,14)(6,13,17,27)(7,19,23,36)(9,16,28,15)(10,25,29,26)(12,20,32,35)(18,34,31,33)(21,24,22,30);(2,4)(5,9,10,27,11,28,29,13)(6,14,15,26,17,8,16,25)(7,20,21,34,23,35,22,33)(12,30,18,19,32,24,31,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303570);