#434721

order	512
nilpotency class	3
isoclinism	isoclinic class of #418863
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1070 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)(5,6,10,15,11,7,16,14)(8,19,23,12,22,29,21,17)(9,24,25,18,26,30,27,13)(20,31,35,36,34,28,33,32);(5,7)(6,11)(8,12)(9,24)(10,14)(13,27)(15,16)(17,22)(18,25)(19,21)(20,32)(23,29)(26,30)(28,33)(31,35)(34,36);(5,8,11,22)(6,12)(7,17)(9,20,26,34)(10,21,16,23)(13,28)(14,29)(15,19)(18,31)(24,36)(25,33,27,35)(30,32);(2,4)(5,9,11,26)(6,13,7,18)(8,20,22,34)(10,27,16,25)(12,28,17,31)(14,24,15,30)(19,32,29,36)(21,35,23,33)

Polycyclic presentation:

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

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, 341917);