#466059

order	512
nilpotency class	3
isoclinism	isoclinic class of #455828
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
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

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:

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

Polycyclic presentation:

2^9|b^2=fg,c^2=ei,d^2=f,e^2=i,f^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,h^b=hi,d^c=de,h^c=hi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 373255);