#147255

order	512
nilpotency class	4
isoclinism	isoclinic class of #2192
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1862
prime radical	C₂⁴	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

510 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54451);