#392698

order	512
nilpotency class	3
isoclinism	isoclinic class of #196248
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35149	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3853 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)(5,6,10,15,11,16,9,14)(7,17,21,28,20,27,19,12)(8,22,23,30,25,13,24,29)(18,31,32,26,34,36,33,35);(5,7)(6,12,16,28)(8,18)(9,19)(10,21)(11,20)(13,26,22,35)(14,27,15,17)(23,32)(24,33)(25,34)(29,36,30,31);(5,8)(6,13)(7,18)(9,23)(10,24)(11,25)(12,26)(14,29)(15,30)(16,22)(17,31)(19,32)(20,34)(21,33)(27,36)(28,35);(2,4)(5,9,11,10)(6,14,16,15)(7,19,20,21)(8,23,25,24)(12,27,28,17)(13,29,22,30)(18,32,34,33)(26,36,35,31)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,f^b=fi,e^c=ei,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235198, #72235212, #72251731, …
maximal normal subgroup of	#72235198, #72235212, #72251731, …
maximal quotient of	#72235858

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299894);