#478304

order	512
nilpotency class	3
isoclinism	isoclinic class of #476613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
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

806 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)(5,6,9,13,11,16,10,15)(7,17,20,12,21,27,18,28)(8,14,24,29,25,30,23,31)(19,32,34,26,22,35,33,36);(5,7,9,20,11,21,10,18)(6,12,15,17,16,28,13,27)(8,22,23,34,25,19,24,33)(14,26,29,35,30,36,31,32);(6,13)(7,18)(8,23)(9,10)(14,30)(15,16)(17,27)(19,22)(20,21)(24,25)(26,35)(32,36);(2,4)(5,8)(6,14)(7,19)(9,24)(10,23)(11,25)(12,26)(13,29)(15,31)(16,30)(17,32)(18,33)(20,34)(21,22)(27,35)(28,36)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385500);