#410693

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2482
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1036 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=h,c^2=fg,d^2=f,e^2=i,h^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,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 317889);