#391967

order	512
nilpotency class	3
isoclinism	isoclinic class of #32870
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

875 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:

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

Polycyclic presentation:

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

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, 299163);