#146793

order	512
nilpotency class	4
isoclinism	isoclinic class of #12952
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1864
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

938 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2)(3,4)(5,6,11,16,12,17,33,18)(7,19,22,52,24,41,23,47)(8,25,28,61,29,62,39,48)(9,30,34,15,10,35,38,49)(13,31,42,59,43,36,50,21)(14,32,44,27,45,37,46,26)(20,53,55,68,57,40,56,67)(51,58,66,54,60,63,65,64);(2,4)(5,7)(6,13,17,43)(8,20)(9,31)(10,36)(11,22)(12,24)(14,40,45,53)(15,47,49,52)(16,42,18,50)(19,30,41,35)(21,38)(23,33)(25,60,62,51)(26,63)(27,58)(28,55)(29,57)(32,64)(34,59)(37,54)(39,56)(44,67,46,68)(48,66,61,65);(5,8,12,29)(6,14,17,45)(7,20,24,57)(9,32,10,37)(11,39,33,28)(13,40,43,53)(15,48,49,61)(16,46,18,44)(19,51,41,60)(21,58,59,63)(22,56,23,55)(25,35,62,30)(26,38,27,34)(31,64,36,54)(42,68,50,67)(47,66,52,65)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=hi,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gi,h^a=hi,d^b=dhi,e^b=eh,h^b=hi,d^c=dh,e^c=eh,f^c=fi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281717, #72282032, #72282164
maximal normal subgroup of	#72281717, #72282032, #72282164

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53989);