#438233

order	512
nilpotency class	3
isoclinism	isoclinic class of #422715
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1001 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)(7,17)(8,10)(9,23)(11,15)(12,16)(13,19)(14,25)(18,33)(20,29)(21,32)(22,27)(24,30)(26,31)(28,35)(34,36);(5,7,10,20,12,21,11,19)(6,13,8,17,16,29,15,32)(9,24,25,18,27,34,26,35)(14,30,22,33,31,36,23,28);(5,8,10,6,12,15,11,16)(7,13,20,32,21,29,19,17)(9,22,25,14,27,23,26,31)(18,28,34,36,35,33,24,30);(2,4)(5,9,10,25,12,27,11,26)(6,14,15,23,16,31,8,22)(7,18,19,24,21,35,20,34)(13,28,17,30,29,33,32,36)

Polycyclic presentation:

2^9|b^2=g,c^2=fi,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,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,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, 345429);