#385871

order	512
nilpotency class	3
isoclinism	isoclinic class of #213752
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35150	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1476 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293067);